On a biased edge isoperimetric inequality for the discrete cube
aa r X i v : . [ m a t h . C O ] M a r ON A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THEDISCRETE CUBE
DAVID ELLIS, NATHAN KELLER, AND NOAM LIFSHITZ
Abstract.
The ‘full’ edge isoperimetric inequality for the discrete cube { , } n (due to Harper, Lindsey, Berstein and Hart) specifies the minimum size of theedge boundary ∂A of a set A ⊂ { , } n , as function of | A | . A weaker (butmore widely-used) lower bound is | ∂A | ≥ | A | log(2 n / | A | ) , where equality holdswhenever A is a subcube. In 2011, the first author obtained a sharp ‘stability’version of the latter result, proving that if | ∂A | ≤ | A | (log(2 n / | A | ) + ǫ ) , thenthere exists a subcube C such that | A ∆ C | / | A | = O ( ǫ/ log(1 /ǫ )) .The ‘weak’ version of the edge isoperimetric inequality has the followingwell-known generalization for the ‘ p -biased’ measure µ p on the discrete cube:if p ≤ / , or if < p < and A is monotone increasing, then pµ p ( ∂A ) ≥ µ p ( A ) log p ( µ p ( A )) .In this paper, we prove a sharp stability version of the latter result, whichgeneralizes the aforementioned result of the first author. Namely, we provethat if pµ p ( ∂A ) ≤ µ p ( A )(log p ( µ p ( A )) + ǫ ) , then there exists a subcube C suchthat µ p ( A ∆ C ) /µ p ( A ) = O ( ǫ ′ / log(1 /ǫ ′ )) , where ǫ ′ := ǫ ln(1 /p ) . This resultis a central component in recent work of the authors proving sharp stabilityversions of a number of Erdős-Ko-Rado type theorems in extremal combina-torics, including the seminal ‘complete intersection theorem’ of Ahlswede andKhachatrian.In addition, we prove a biased-measure analogue of the ‘full’ edge isoperi-metric inequality, for monotone increasing sets, and we observe that such ananalogue does not hold for arbitrary sets, hence answering a question of Kalai.We use this result to give a new proof of the ‘full’ edge isoperimetric inequality,one relying on the Kruskal-Katona theorem. Introduction
Isoperimetric inequalities are of ancient interest in mathematics. In general,an isoperimetric inequality gives a lower bound on the ‘boundary-size’ of a setof a given ‘size’, where the exact meaning of these words varies according to theproblem. In the last fifty years, there has been a great deal of interest in discrete isoperimetric inequalities. These deal with the ‘boundary’ of a set A of vertices in agraph G = ( V, E ) – either the edge boundary ∂A , which consists of the set of edgesof G that join a vertex in A to a vertex in V \ A , or the vertex boundary b ( A ) , whichconsists of the set of vertices of V \ A that are adjacent to a vertex in A .1.1. The edge isoperimetric inequality for the discrete cube, and somestability versions thereof.
A specific discrete isoperimetric problem which at-tracted much interest due to its numerous applications is the edge isoperimetricproblem for the n -dimensional discrete cube, Q n . This is the graph with vertex-set Date : 3rd February 2017.The research of N.K. was supported by the Israel Science Foundation (grant no. 402/13), theBinational US-Israel Science Foundation (grant no. 2014290), and by the Alon Fellowship.
1N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 2 { , } n , where two 0-1 vectors are adjacent if they differ in exactly one coordi-nate. The edge isoperimetric problem for Q n was solved by Harper [19], Lindsey[31], Bernstein [3], and Hart [20]. Let us describe the solution. We may identify { , } n with the power-set P ([ n ]) of [ n ] := { , , . . . , n } , by identifying a 0-1 vector ( x , . . . , x n ) with the set { i ∈ [ n ] : x i = 1 } . We can then view Q n as the graphwith vertex set P ([ n ]) , where two sets S, T ⊂ [ n ] are adjacent if | S ∆ T | = 1 . The lexicographic ordering on P ([ n ]) is defined by S > T iff min( S ∆ T ) ∈ S . If m ∈ [2 n ] ,the initial segment of the lexicographic ordering on P ([ n ]) of size m (or, in short,the lexicographic family of size m ) is simply the m largest elements of P ([ n ]) withrespect to the lexicographic ordering. Harper, Bernstein, Lindsey and Hart provedthe following. Theorem 1.1 (The ‘full’ edge isoperimetric inequality for Q n ) . If F ⊂ P ([ n ]) then | ∂ F| ≥ | ∂ L| , where L ⊂ P ([ n ]) is the initial segment of the lexicographic orderingof size |F| . A weaker, but more convenient (and, as a result, more widely-used) lower bound,is the following:
Corollary 1.2 (The weak edge isoperimetric inequality for Q n ) . If F ⊂ P ([ n ]) then (1.1) | ∂ F| ≥ |F| log (2 n / |F| ) . Equality holds in (1.1) iff F is a subcube, so (1.1) is sharp only when |F| is apower of 2.When an isoperimetric inequality is sharp, and the extremal sets are known, itis natural to ask whether the inequality is also ‘stable’ — i.e., if a set has boundaryof size ‘close’ to the minimum, must that set be ‘close in structure’ to an extremalset?For Corollary 1.2, this problem was studied in several works. Using a Fourier-analytic argument, Friedgut, Kalai and Naor [18] obtained a stability result forsets of size n − , showing that if F ⊂ P ([ n ]) with |F| = 2 n − satisfies | ∂ F| ≤ (1 + ǫ )2 n − , then |F ∆ C| / n = O ( ǫ ) for some codimension-1 subcube C . (Thedependence upon ǫ here is almost sharp, viz., sharp up to a factor of Θ(log(1 /ǫ )) ).Bollobás, Leader and Riordan (unpublished) proved an analogous result for |F| ∈{ n − , n − } , also using a Fourier-analytic argument. Samorodnitsky [34] used aresult of Keevash [27] on the structure of r -uniform hypergraphs with small shadows,to prove a stability result for all F ⊂ P ([ n ]) with log |F| ∈ N (i.e., all sizesfor which Corollary 1.2 is tight), under the rather strong condition | ∂ F| ≤ (1 + O (1 /n )) | ∂ L| . In [6], the first author proved the following stability result (whichimplies the above results), using a recursive approach and an inequality of Talagrand[35] (which was proved via Fourier analysis). Theorem 1.3 ([6]) . There exists an absolute constant c > such that the followingholds. Let ≤ δ < c . If F ⊂ P ([ n ]) with |F| = 2 d for some d ∈ N , and |F ∆ C| ≥ δ d for all d -dimensional subcubes C ⊂ P ([ n ]) , then | ∂ F| ≥ | ∂ C| + 2 d δ log (1 /δ ) . As observed in [6], this result is best-possible (except for the condition ≤ δ < c ,which was conjectured to be unnecessary in [6]). N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 3
In [9], we obtain the following stability version of Theorem 1.1, which appliesto families of arbitrary size (not just a power of 2), and which is sharp up to anabsolute constant factor.
Theorem 1.4.
There exists an absolute constant
C > such that the followingholds. If F ⊂ P ([ n ]) and L ⊂ P ([ n ]) is the initial segment of the lexicographicordering of size |F| , then there exists an automorphism σ of Q n such that |F ∆ σ ( L ) | ≤ C ( | ∂ F| − | ∂ L| ) . The proof uses only combinatorial tools, but is much more involved than theproof of Theorem 1.3 in [6].1.2.
Influences of Boolean functions.
An alternative viewpoint on the edgeisoperimetric inequality, which we will use throughout the paper, is via influences of Boolean functions. For a function f : { , } n → { , } , the influence of the i thcoordinate on f is defined by I i [ f ] := Pr x ∈{ , } n [ f ( x ) = f ( x ⊕ e i )] , where x ⊕ e i is obtained from x by flipping the i th coordinate, and the probabilityis taken with respect to the uniform measure on { , } n . The total influence of thefunction is I [ f ] := n X i =1 I i [ f ] . Over the last thirty years, many results have been obtained on the influences ofBoolean functions, and have proved extremely useful in such diverse fields as the-oretical computer science, social choice theory and statistical physics, as well as incombinatorics (see, e.g., the survey [25]).It is easy to see that the total influence of a function f is none other than the sizeof the edge boundary of the set A ( f ) = { x ∈ { , } n : f ( x ) = 1 } , appropriatelynormalised: viz., I [ f ] = | ∂ ( A ( f )) | / n − . Hence, Corollary 1.2 has the followingreformulation in terms of Boolean functions and influences: Proposition 1.5 (The weak edge isoperimetric inequality for Q n – influence ver-sion) . If f : { , } n → { , } is a Boolean function then (1.2) I [ f ] ≥ E [ f ] log (1 / E [ f ]) . Theorem 1.3 can be restated similarly.1.3.
The biased measure on the discrete cube.
For p ∈ [0 , , the p -biasedmeasure on P ([ n ]) is defined by µ ( n ) p ( S ) = p | S | (1 − p ) n −| S | ∀ S ⊂ [ n ] . In other words, we choose a random subset of [ n ] by including each j ∈ [ n ] inde-pendently with probability p . When n is understood, we will omit the superscript ( n ) , writing µ p = µ ( n ) p .The definition of influences with respect to the biased measure is, naturally, I pi [ f ] := Pr x ∼ µ p [ f ( x ) = f ( x ⊕ e i )] , and I p [ f ] := P ni =1 I pi [ f ] . We abuse notation slightly and write µ p ( f ) := E µ p [ f ] .We remark that we may write I p [ f ] = µ p ( ∂A ( f )) , where we define the measure µ p N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 4 on subsets of E ( Q n ) by µ p ( { x, x ⊕ e i } ) = p P j = i x i (1 − p ) n − − P j = i x j . (Note that µ p ( E ( Q n )) = n , so µ p is not a probability measure on E ( Q n ) unless n = 1 .)Many of the applications of influences (e.g., to the study of percolation [2],threshold phenomena in random graphs [4, 16], and hardness of approximation [5])rely upon the use of the biased measure on the discrete cube. As a result, manyof the central results on influences have been generalized to the biased setting (e.g.[15, 17, 21]), and the edge isoperimetric inequality is no exception. The following‘biased’ generalization of Proposition 1.5 is considered folklore (see [22]). Theorem 1.6 (The weak biased edge isoperimetric inequality for Q n ) . If f : { , } n → { , } is a Boolean function, and < p ≤ / , then (1.3) pI p [ f ] ≥ µ p ( f ) log p ( µ p ( f )) . The same statement holds for all p ∈ (0 , if f is monotone increasing. Note that a function f : { , } n → { , } is said to be monotone increasing if f ( x ) ≤ f ( y ) whenever x i ≤ y i for all i ∈ [ n ] . An easy inductive proof of Theorem1.6 is presented in [22].1.4. A stability version of the biased edge isoperimetric inequality.
Thefirst main result of this paper is the following stability version of Theorem 1.6.
Theorem 1.7.
There exist absolute constants c , C > such that the followingholds. Let < p ≤ , and let ǫ ≤ c / ln(1 /p ) . Let f : { , } n → { , } be a Booleanfunction such that pI p [ f ] ≤ µ p ( f ) (cid:0) log p ( µ p ( f )) + ǫ (cid:1) . Then there exists a subcube S ⊂ { , } n such that (1.4) µ p ( f ∆1 S ) ≤ C ǫ ln(1 /p )ln (1 / ( ǫ ln(1 /p ))) µ p ( f ) , where f ∆1 S := { x : f ( x ) = 1 S ( x ) } . If we assume further that f is monotone increasing, then the above theorem can beextended to p > / . Theorem 1.8.
For any η > , there exist C = C ( η ) , c = c ( η ) > such that thefollowing holds. Let < p ≤ − η , and let ǫ ≤ c / ln(1 /p ) . Let f : { , } n → { , } be a monotone increasing Boolean function such that pI p [ f ] ≤ µ p ( f ) (cid:0) log p ( µ p ( f )) + ǫ (cid:1) . Then there exists a monotone increasing subcube S ⊂ { , } n such that (1.5) µ p ( f ∆1 S ) ≤ C ǫ ln(1 /p )ln (1 / ( ǫ ln(1 /p )) µ p ( f ) . (Note subset S ⊂ { , } n is said to be monotone increasing if its indicator function ismonotone increasing. The indicator function of S ⊂ { , } n is the Boolean functionon { , } n taking the value on S and outside S .)As we show in Section 4, Theorems 1.7 and 1.8 are sharp, up to the values of theconstants c , C , and this remains the case even if the subcube in the conclusion ofTheorem 1.8 is allowed to be non-monotone. Moreover, the dependence of c , C on η in Theorem 1.8 cannot be removed — though, for the sake of brevity, we donot attempt to optimise the dependence of these constants on η in our proof. N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 5
The proofs of Theorems 1.7 and 1.8 use induction on n , in a similar way to theproof of Theorem 1.3 in [6], but unlike in previous works, they do not use anyFourier-theoretic tools, relying only upon ‘elementary’ (though intricate) combina-torial and analytic arguments.Theorems 1.7 and 1.8 are crucial tools in a recent work of the authors [8], whichestablishes a general method for leveraging Erdős-Ko-Rado type results in extremalcombinatorics into strong stability versions, without going into the proofs of theoriginal results. This method is used in [8] to obtain sharp (or almost-sharp) sta-bility versions of the Erdős-Ko-Rado theorem itself [11], of the seminal ‘completeintersection theorem’ of Ahlswede and Khachatrian [1], of Frankl’s recent resulton the Erdős matching conjecture [12], of the Ellis-Filmus-Friedgut proof of theSimonovits-Sós conjecture [7], and of various Erdős-Ko-Rado type results on r -wise(cross)- t -intersecting families.Theorem 1.8 is also used in [10] by the first and last authors to obtain sharpupper bounds on the size of the union of several intersecting families of k -elementsubsets of [ n ] , where k ≤ (1 / − o (1)) n , extending results of Frankl and Füredi [14].1.5. A biased version of the ‘full’ edge isoperimetric inequality for mono-tone increasing families.
While the generalization of the ‘weak’ edge isoperi-metric inequality (i.e., Corollary 1.2) to the biased measure has been known fora long time, such a generalization of the ‘full’ edge isoperimetric inequality (i.e.,Theorem 1.1) was hitherto unknown. In his talk at the 7th European Congressof Mathematicians [24], Kalai asked whether there is a natural generalization ofTheorem 1.1 to the measure µ p for p < / .We answer Kalai’s question in the affirmative by showing that the most naturalsuch generalization does not hold for arbitrary families, but does hold (even for p > / ) under the additional assumption that the family is monotone increasing.(We say a family F ⊂ P ([ n ]) is monotone increasing if ( S ∈ F , S ⊂ T ) ⇒ T ∈ F .)In order to present our result, we first define the appropriate generalizationof lexicographic families for the biased-measure setting. Note that while in theuniform measure ( p = 1 / ) case, for any F ⊂ P ([ n ]) there exists a lexicographicfamily L ⊂ P ([ n ]) with the same measure as F , this does not hold in general for p = 1 / . However, the situation can be remedied by passing to subsets of theCantor space P ( N ) . We let Σ be the σ -algebra on P ( N ) generated by ∪ n ∈ N P ([ n ]) ,and for each p ∈ (0 , , we let µ ( N ) p be the natural p -biased measure on ( P ( N ) , Σ) (the unique measure that ‘projects’ to the measure µ ( n ) p on P ([ n ]) , for each n ∈ N ).By analogy with subsets of [ n ] , if F ∈ Σ and i ∈ N we define the i th influence of F w.r.t. µ ( N ) p by I pi [ F ] := Pr S ∼ µ ( N ) p [ F ∩ {
S, S ∆ { i }}| = 1] and the total influence of F w.r.t. µ ( N ) p by I p [ F ] = P ∞ i =1 I pi [ F ] .Just as for subsets of [ n ] , the lexicographic ordering on P ( N ) is defined by S > T iff min( S ∆ T ) ∈ S . For each λ ∈ [0 , , we let L λ ⊂ P ( N ) be the unique initial seg-ment of the lexicographic ordering on P ( N ) with µ ( N )1 / ( L λ ) = λ . (It is easily checkedthat initial segments of the lexicographic ordering on P ( N ) are Σ -measurable.)Moreover, the function f p : λ µ ( N ) p ( L λ ) is continuous and monotone increasing,for each p ∈ (0 , , with f p (0) = 0 and f p (1) = 1 . Hence, by the intermediate N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 6 value theorem, for any p ∈ (0 , and any x ∈ [0 , , there exists λ ∈ [0 , such that µ ( N ) p ( L λ ) = x . In particular, for each n ∈ N and each F ⊂ P ([ n ]) , there exists λ ∈ [0 , such that µ ( N ) p ( L λ ) = µ ( n ) p ( F ) , where µ ( n ) p denotes the p -biased measureon P ([ n ]) . We prove this family L λ has total influence no larger than that of F : Theorem 1.9.
Let p ∈ (0 , , and let F ⊂ P ([ n ]) be a monotone increasing family.Let λ ∈ [0 , be such that µ ( N ) p ( F ) = µ ( n ) p ( L λ ) . Then I p [ F ] ≥ I p [ L λ ] . (Here, I p [ F ] is defined in terms of the p -biased measure on P ([ n ]) , whereas I p [ L λ ] is defined interms of the p -biased measure on ( P ( N ) , Σ) .) Our proof uses the Kruskal-Katona theorem [26, 29], the Margulis-Russo Lemma[32, 33], and some additional analytic and combinatorial arguments.In fact, Theorem 1.1 (the ‘full’ edge-isoperimetric inequality of Harper, Bern-stein, Lindsey and Hart) follows quickly from Theorem 1.9, via a monotonizationargument, so our proof of Theorem 1.9 provides a new proof of Theorem 1.1, via theKruskal-Katona theorem. This may be of independent interest, and may be some-what surprising, as the Kruskal-Katona theorem is more immediately connected tothe vertex-boundary of an increasing family, than to its edge-boundary.We remark that the assertion of Theorem 1.9 is false for arbitrary (i.e., non-monotone) functions, for each value of p = 1 / . Indeed, it is easy to check thatfor each p ∈ (0 , \ { } , the ‘antidictatorship’ A = { S ⊂ [ n ] : 1 / ∈ S } has I p [ A ] = 1 < I p [ L λ ] , where λ is such that µ p ( L λ ) = 1 − p (= µ p ( A )) . (See Remark5.12.)1.6. Organization of the paper.
In Section 2, we outline some notation andpresent an inductive proof of Theorem 1.6, some of whose ideas and componentswe will use in the sequel. In Section 3 (the longest part of the paper), we proveTheorems 1.7 and 1.8. In Section 4, we give examples showing that Theorems 1.7and 1.8 are sharp (in a certain sense). In Section 5, we prove Theorem 1.9 andshow how to use it to deduce Theorem 1.1. We conclude the paper with some openproblems in Section 6.2.
An inductive proof of Theorem 1.6
In this section, we outline some notation and terminology, and present a simpleinductive proof of Theorem 1.6; components and ideas from this proof will be usedin the proofs of Theorems 1.7 and 1.8.2.1.
Notation and terminology.
When the ‘bias’ p (of the measure µ p ) is clearfrom the context (including throughout Sections 2 and 3), we will sometimes omitit from our notation, i.e. we will sometimes write µ ( f ) := µ p ( f ) and I [ f ] := I p [ f ] .Moreover, when the Boolean function f is clear from the context, we will sometimesomit it from our notation, i.e. we will sometimes write µ := µ ( f ) , I := I [ f ] and I i := I i [ f ] . If S ⊂ { , } n , we write S for its indicator function, i.e. the Booleanfunction on { , } n taking the value on S and outside S . A dictatorship is aBoolean function f : { , } n → { , } of the form f = 1 { x j =1 } for some j ∈ [ n ] ; an antidictatorship is one of the form f = 1 { x j =0 } . Abusing notation slightly, we willsometimes identify a family F ⊂ P ([ n ]) with the corresponding indicator function { x ∈{ , } n : { i ∈ [ n ]: x i =1 }∈F} . N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 7 A subcube of { , } n is a set of the form { x ∈ { , } n : x i = a i ∀ i ∈ F } , where F ⊂ [ n ] and a i ∈ { , } for all i ∈ F ; F is called the set of fixed coordinates of thesubcube.We use the convention p (0) = 0 (for all p ∈ (0 , ); this turns x x log p ( x ) into a continuous function on [0 , . If S and T are sets, we write S ⊂ T if S is a(not necessarily proper) subset of T .If f : { , } n → { , } and i ∈ [ n ] , we define the function f i → : { , } [ n ] \{ i } →{ , } by f i ( y ) = f ( x ) , where x i = 0 and x j = y j for all j ∈ [ n ] \ { i } . In otherwords, f i → is the restriction of f to the lower half-cube { x ∈ { , } n : x i = 0 } . Wedefine f i → similarly. For brevity, we will often write µ − i = µ − i ( f ) := µ p ( f i → ) ,µ + i = µ + i ( f ) := µ p ( f i → ) ,I − i = I − i [ f ] := I p [ f i → ] ,I + i = I + i [ f ] := I p [ f i → ] . Note that(2.1) pµ + i ( f ) + (1 − p ) µ − i ( f ) = µ ( f ) and that(2.2) I [ f ] = I i [ f ] + pI + i [ f ] + (1 − p ) I − i [ f ] . A proof of Theorem 1.6.
The proof uses induction on n together withequations (2.1) and (2.2), and the following technical lemma. Lemma 2.1.
Let p ∈ (0 , , and let F, G, H : [0 , × [0 , → [0 , ∞ ) be the functionsdefined by F ( x, y ) = px log p x + (1 − p ) y log p y + px − py,G ( x, y ) = ( px + (1 − p ) y ) log p (( px + (1 − p ) y )) ,H ( x, y ) = px log p x + (1 − p ) y log p y + py − px. (1) If x ≥ y ≥ , then F ( x, y ) ≥ G ( x, y ) . (2) If y ≥ x ≥ and p ≤ , then H ( x, y ) ≥ G ( x, y ) .Proof of Lemma 2.1. Clearly, for all y ≥ we have F ( y, y ) = G ( y, y ) = H ( y, y ) ,and for all x, y ≥ , we have ∂F∂x = p log p x + p ln p + p = p log p ( px ) + p ln p ,∂G∂x = p log p ( px + (1 − p ) y ) + p ln p ,∂H∂y = (1 − p ) log p y + p + 1 − p ln p (2.3) = (1 − p ) log p ((1 − p ) y ) − (1 − p ) log p (1 − p ) + p + 1 − p ln p ,∂G∂y = (1 − p ) log p ( px + (1 − p ) y ) + 1 − p ln p . N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 8
Clearly, we have ∂F ( x,y ) ∂x ≥ ∂G ( x,y ) ∂x for all x, y ≥ , and therefore F ( x, y ) ≥ G ( x, y ) for all x ≥ y ≥ , proving (1). We assert that similarly, ∂H ( x,y ) ∂y ≥ ∂G ( x,y ) ∂y for all x, y ≥ , if p ≤ / . (This will imply (2).) Indeed, ∂H∂y = (1 − p ) log p ((1 − p ) y ) − (1 − p ) log p (1 − p ) + p + 1 − p ln p ≥ ∂G∂y + p − (1 − p ) log p (1 − p ) . Hence, it suffices to prove the following.
Claim 2.2.
Define K : (0 , → R ; K ( p ) = p − (1 − p ) log p (1 − p ) . Then K ( p ) > for all p ∈ (0 , ) , K (1 /
2) = 0 and K ( p ) < for all p ∈ (1 / , .Proof of Claim 2.2. Clearly, we have K (1 /
2) = 0 . It suffices to show that α ( p ) := K ( p ) ln(1 /p ) = − p ln p + (1 − p ) ln(1 − p ) is positive for all p ∈ (0 , / , since α (1 − p ) = − α ( p ) for all p ∈ (0 , . Notethat α ( x ) → as x → and that α ( x ) → as x → , so we may extend α to acontinuous function on [0 , by defining α (0) = α (1) = 0 .We have α ′ ( x ) = − ln x − ln(1 − x ) − . Suppose for a contradiction that α has a zero in (0 , / . Then, since and / are also zeros of α , α would have at least two stationary points in (0 , / . Thiscannot occur, because α ′ ( x ) = 0 implies x (1 − x ) = e − , which has at most onesolution in (0 , / , since if x is a solution then − x is also solution, and anyquadratic equation has at most two solutions. Hence, α has no zeros in (0 , / .Since α ′ ( x ) → ∞ as x → , we must have α ( x ) > for all x ∈ (0 , / , asrequired. (cid:3) This completes the proof of Lemma 2.1. (cid:3)
We can now prove Theorem 1.6.
Proof of Theorem 1.6.
It is easy to check that the theorem holds for n = 1 . Let n ≥ , and suppose the statement of the theorem holds when n is replaced by n − .Let f : { , } n → { , } . Choose any i ∈ [ n ] . We split into two cases.Case (a) µ − i ≤ µ + i .Applying the induction hypothesis to the functions f i → and f i → , and usingthe fact that I i [ f ] ≥ µ + i − µ − i , we obtain pI = (1 − p ) pI − i [ f ] + p I + i [ f ] + pI i [ f ] ≥ (1 − p ) µ − i log p ( µ − i ) + pµ + i log p ( µ + i ) + p (cid:0) µ + i − µ − i (cid:1) = F (cid:0) µ + i , µ − i (cid:1) ≥ G (cid:0) µ + i , µ − i (cid:1) = µ log p ( µ ) , where F and G are as defined in Lemma 2.1.Case (b) µ − i ≥ µ + i . N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 9
The proof in this case is similar: applying the induction hypothesis to the func-tions f i → and f i → , and using the fact that I i [ f ] ≥ µ − i − µ + i , we obtain pI = (1 − p ) pI − i [ f ] + p I + i [ f ] + pI i [ f ] ≥ (1 − p ) µ − i log p ( µ − i ) + pµ + i log p ( µ + i ) + p (cid:0) µ − i − µ + i (cid:1) = H (cid:0) µ + , µ − i (cid:1) ≥ G (cid:0) µ + i , µ − i (cid:1) = µ log p ( µ ) , using the fact that p ≤ / . (cid:3) We remark that the above proof shows that if f is monotone increasing, thenthe statement of Theorem 1.6 holds for all p ∈ (0 , . (Indeed, if f is monotoneincreasing, then µ − i ≤ µ + i for all i ∈ [ n ] , so the assumption p ≤ / is not required.)3. Proofs of the ‘biased’ isoperimetric stability theorems
In this section, we prove Theorems 1.7 and 1.8. As the proofs of the two theoremsfollow the same strategy, we present them in parallel.The proof of Theorem 1.7 (and similarly, of Theorem 1.8) consists of five steps.Assume that f satisfies the assumptions of the theorem.(1) We show that for each i ∈ [ n ] , either I i [ f ] is small or else min (cid:8) µ − i , µ + i (cid:9) is ‘somewhat’ small. In other words, the influences of f are similar to theinfluences of a subcube.(2) We show that µ must be either very close to 1 or ‘fairly’ small, i.e., boundedaway from 1 by a constant. (In the proof of Theorem 1.8, the constant maydepend on η .)(3) We show that unless µ is very close to 1, there exists i ∈ [ n ] such that I i [ f ] is large. This implies that min { µ − i , µ + i } is ‘somewhat’ small.(4) We prove two ‘bootstrapping’ lemmas saying that if µ − i is ‘somewhat’ small,then it must be ‘very’ small, and that if µ + i is ‘somewhat’ small, then itmust be ‘very’ small. This implies that f is ‘very’ close to being containedin a dictatorship or an antidictatorship.(5) Finally, we prove each theorem by induction on n .From now on, we let f : { , } n → { , } such that pI p [ f ] ≤ µ p ( f )(log p ( µ p ( f )) + ǫ ) .By reducing ǫ if necessary, we may assume that pI p [ f ] = µ p ( f )(log p ( µ p ( f )) + ǫ ) ,i.e., using the more compact notation outlined above, pI [ f ] = µ (log p ( µ ) + ǫ ) .3.1. Relations between the influences of f and the influences of its re-strictions f i → , f i → . We define ǫ − i , ǫ + i by pI − i = µ − i (cid:0) log p ( µ − i ) + ǫ − i (cid:1) , pI + i = µ + i (cid:0) log p ( µ + i ) + ǫ + i (cid:1) . Note that Theorem 1.6 implies that ǫ − i , ǫ + i ≥ . We define the functions F, G, H, K as in the proof of Theorem 1.6.We would now like to express the fact that I [ f ] is small in terms of ǫ − i , ǫ + i , µ − i , µ + i .For each i ∈ [ n ] such that µ − i ≤ µ + i , we have µ (log p ( µ ) + ǫ ) = pI [ f ] = (1 − p ) pI − i + p I + i + pI i [ f ]= (1 − p ) µ − i (cid:0) log p ( µ − i ) + ǫ − i (cid:1) + pµ + i (cid:0) log p ( µ + i ) + ǫ + i (cid:1) + p ( µ + i − µ − i ) + p ( I i [ f ] − µ + i + µ − i ); (3.1) N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 10 rearranging (3.1) gives ǫ ′ i : = µǫ − pµ + i ǫ + i − (1 − p ) µ − i ǫ − i = pµ + i ( ǫ − ǫ + i ) + (1 − p ) µ − i ( ǫ − ǫ − i )= F (cid:0) µ + i , µ − i (cid:1) − G (cid:0) µ + i , µ − i (cid:1) + p (cid:0) I i [ f ] − ( µ + i − µ − i ) (cid:1) . (3.2)Similarly, for each i ∈ [ n ] such that µ − i ≥ µ + i , we have ǫ ′ i := µǫ − pµ + i ǫ + i − (1 − p ) µ − i ǫ − i = pµ + i ( ǫ − ǫ + i ) + (1 − p ) µ − i ( ǫ − ǫ − i )= H (cid:0) µ + i , µ − i (cid:1) − G (cid:0) µ + i , µ − i (cid:1) + p (cid:0) I i [ f ] − ( µ − i − µ + i ) (cid:1) . (3.3)This allows us to deduce two facts about the structure of f . • By Lemma 2.1, we have ǫ ′ i ≥ for all i ∈ [ n ] . This implies that either ǫ + i ≤ ǫ or ǫ − i ≤ ǫ . Together with the induction hypothesis, this will imply(in Section 3.6) that either f i → or f i → is structurally close to a subcube. • F (cid:0) µ + i , µ − i (cid:1) − G (cid:0) µ + i , µ − i (cid:1) (resp. H (cid:0) µ + i , µ − i (cid:1) − G (cid:0) µ + i , µ − i (cid:1) ) is small whenever µ + i ≥ µ − i (resp. µ − i ≥ µ + i ). Note that the proof of Lemma 2.1 shows thatwhenever µ + i ≥ µ − i (resp. µ − i ≥ µ + i ) then F (cid:0) µ + i , µ − i (cid:1) (resp. H (cid:0) µ + i , µ − i (cid:1) )is equal to G (cid:0) µ + i , µ − i (cid:1) only if µ + i = µ − i or µ − i = 0 . We will later show(in Claims 3.2-3.4) that if F (cid:0) µ + i , µ − i (cid:1) (resp. H (cid:0) µ + i , µ − i (cid:1) ) is approximatelyequal to G (cid:0) µ + i , µ − i (cid:1) , then either min (cid:8) µ − i , µ + i (cid:9) is small or else I i [ f ] is small.The following lemma will be used to relate µ + i and µ − i to F ( µ + i , µ − i ) − G ( µ + i , µ − i ) (or to H ( µ + i , µ − i ) − G ( µ + i , µ − i ) ), in a more convenient way. Lemma 3.1. If < p < and x ≥ y ≥ , then F ( x, y ) − G ( x, y ) ≥ p ( x − y ) log p (cid:18) pxpx + (1 − p ) y (cid:19) . If < p ≤ and y ≥ x ≥ , then H ( x, y ) − G ( x, y ) ≥ (1 − p ) ( y − x ) log p (cid:18) (1 − p ) ypx + (1 − p ) y (cid:19) . If < p ≤ e − and y ≥ x ≥ , then H ( x, y ) − G ( x, y ) ≥ p ( y − x ) . Proof.
We show that ∂∂u ( F ( u, y ) − G ( u, y )) (cid:12)(cid:12)(cid:12)(cid:12) u = t ≥ p log p (cid:18) pxpx + (1 − p ) y (cid:19) ∀ y ≤ t ≤ x, < p < , (3.4) ∂∂u ( H ( x, u ) − G ( x, u )) (cid:12)(cid:12)(cid:12)(cid:12) u = t ≥ (1 − p ) log p (cid:18) (1 − p ) ypx + (1 − p ) y (cid:19) ∀ x ≤ t ≤ y, < p ≤ / , (3.5) ∂∂u ( H ( x, u ) − G ( x, u )) (cid:12)(cid:12)(cid:12)(cid:12) u = t ≥ p, ∀ x ≤ t ≤ y, < p ≤ e − . (3.6)These inequalities will complete the proof of the lemma, by the Fundamental The-orem of Calculus. N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 11
Using (2.3), we have ∂∂u ( F ( u, y ) − G ( u, y )) (cid:12)(cid:12)(cid:12)(cid:12) u = t = p log p ( pt ) + p ln ( p ) − p log p ((1 − p ) y + pt ) − p ln ( p )= p log p (cid:18) pt (1 − p ) y + pt (cid:19) ≥ p log p (cid:18) px (1 − p ) y + px (cid:19) , proving (3.4). Similarly, if p ≤ / and x ≤ t ≤ y , then ∂∂u ( H ( x, u ) − G ( x, u )) (cid:12)(cid:12)(cid:12)(cid:12) u = t = (1 − p ) log p (cid:18) (1 − p ) tpx + (1 − p ) t (cid:19) + K ( p ) ≥ (1 − p ) log p (cid:18) (1 − p ) ypx + (1 − p ) y (cid:19) + K ( p ) ≥ (1 − p ) log p (cid:18) (1 − p ) ypx + (1 − p ) y (cid:19) , proving (3.5). It is easy to check that for all p ≤ e − , we have K ( p ) ≥ p . Hence,if x ≤ t ≤ y and < p ≤ e − , then ∂∂u ( H ( x, u ) − G ( x, u )) (cid:12)(cid:12)(cid:12)(cid:12) u = t ≥ K ( p ) ≥ p , proving (3.6). (cid:3) Either I i [ f ] is small, or min (cid:8) µ − i , µ + i (cid:9) is small. We now show that theinfluences of f are similar to the influences of a subcube. Note that if f = 1 S fora subcube S = { x ∈ { , } n : x i = a i ∀ i ∈ T } , where T ⊂ [ n ] and a i ∈ { , } forall i ∈ T , then min (cid:8) µ − i , µ + i (cid:9) = 0 for each i ∈ T , and I i [ f ] = 0 for each i / ∈ T . Weprove that an approximate version of this statement holds, under our hypotheses.We start with the simplest case, which is ζ < p ≤ for some ζ > . Claim 3.2.
Let ζ > . There exists C = C ( ζ ) > such that if ζ ≤ p ≤ / , thenfor each i ∈ [ n ] , one of the following holds. Case (1):
We have I i [ f ] ≤ C ǫ ′ i , and min (cid:8) µ − i , µ + i (cid:9) ≥ (1 − C ǫ ) µ . Case (2):
We have min (cid:8) µ − i , µ + i (cid:9) ≤ C ǫ ′ i , and I i [ f ] ≥ (1 − C ǫ ) µ . We remark that in Claim 3.2, it is necessary that C depend on ζ ; this is evi-denced e.g. by the function f = 1 B in Section 4, with t = 1 , s = 3 and i = 2 . Proof of Claim 3.2.
By Lemma 3.1 and (3.2), if µ − i ≤ µ + i then p (cid:0) µ + i − µ − i (cid:1) log p (cid:18) pµ + i µ (cid:19) ≤ F (cid:0) µ + i , µ − i (cid:1) − G (cid:0) µ + i , µ − i (cid:1) ≤ ǫ ′ i − pI i [ f ] − pµ − i + pµ + i . (3.7)By Lemma 3.1 and (3.3), if µ − i ≥ µ + i then (1 − p ) (cid:0) µ − i − µ + i (cid:1) log p (cid:18) (1 − p ) µ − i µ (cid:19) ≤ H (cid:0) µ + i , µ − i (cid:1) − G (cid:0) µ + i , µ − i (cid:1) ≤ ǫ ′ i − pI i [ f ] − pµ + i + pµ − i . (3.8)Since the right-hand sides of (3.7) and (3.8) are non-negative, we have(3.9) I i [ f ] − (cid:12)(cid:12) µ + i − µ − i (cid:12)(cid:12) ≤ p ǫ ′ i ≤ ζ ǫ ′ i . We now split into two cases.
N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 12
Case (a): min (cid:8) µ − i , µ + i (cid:9) ≥ µ .In this case, we have p log p (cid:18) pµ + i µ (cid:19) = Ω ζ (1) , (1 − p ) log p (cid:18) (1 − p ) µ − i µ (cid:19) = Ω ζ (1) , so (cid:12)(cid:12) µ + i − µ − i (cid:12)(cid:12) = O ζ ( ǫ ′ i ) , by (3.7) and (3.8). Equation (3.9) now implies that I i [ f ] = O ζ ( ǫ ′ i ) . Therefore, min (cid:8) µ − i , µ + i (cid:9) ≥ µ − I i [ f ] = µ − O ζ ( ǫ ′ i ) = µ − O ζ ( ǫµ ) = µ (1 − O ζ ( ǫ )) . (Note that,by the definition of ǫ ′ i in (3.2) and(3.3), we always have ǫ ′ i ≤ ǫµ .) Hence, Case (1)of the claim occurs.Case (b): min (cid:8) µ − i , µ + i (cid:9) ≤ µ .Firstly, suppose in addition that µ − i ≤ µ + i , so that µ − i ≤ µ/ . Then p ( µ + i − µ − i ) ≥ p ( µ − µ − i ) ≥ pµ/ ≥ ζµ/ ζ ( µ ) , so (3.7) implies that log p (cid:18) pµ + i µ (cid:19) = O ζ ( ǫ ′ i /µ ) . Hence, ln (cid:16) µpµ + i (cid:17) = O ζ ( ǫ ′ i /µ ) , and therefore − p ) µ − i pµ + i = µpµ + i = exp (cid:16) O ζ ( ǫ ′ i /µ ) (cid:17) = 1 + O ζ ( ǫ ′ i /µ ) . Therefore, µ − i = O ζ ( ǫ ′ i ) pµ + i (1 − p ) µ = O ζ ( ǫ ′ i ) . We now have I i [ f ] ≥ µ − µ − i = µ − O ζ ( ǫ ′ i ) = µ − O ζ ( ǫµ ) = (1 − O ζ ( ǫ )) µ . Hence, Case (2) of the claim occurs.Secondly, suppose in addition that µ + i ≤ µ − i , so that µ + i ≤ µ/ . Then we have (1 − p )( µ − i − µ + i ) = Ω ( µ ) , so (3.8) implies that log p (cid:18) (1 − p ) µ − i µ (cid:19) = O ( ǫ ′ i /µ ) . Hence, ln (cid:16) µ (1 − p ) µ − i (cid:17) = O ζ ( ǫ ′ i /µ ) , and therefore pµ + i (1 − p ) µ − i = µ (1 − p ) µ − i = exp (cid:16) O ζ ( ǫ ′ i /µ ) (cid:17) = 1 + O ζ ( ǫ ′ i /µ ) . Therefore, µ + i = O ζ ( ǫ ′ i ) (1 − p ) µ − i pµ = O ζ ( ǫ ′ i ) . It follows that I i [ f ] ≥ (1 − O ζ ( ǫ )) µ , soagain, Case (2) of the claim must occur. (cid:3) We now prove a version of Claim 3.2 for monotone increasing f and for all p bounded away from 1. The idea of the proof is the same, but the details areslightly messier, mainly because p is no longer bounded away from . Claim 3.3.
For any η > , there exists C = C ( η ) > such that the followingholds. Suppose that f is monotone increasing and that < p ≤ − η . Let i ∈ [ n ] .Then one of the following must occur. Case (1):
We have pI i [ f ] ≤ C ǫ ′ i ln(1 /p ) , and µ − i ≥ (1 − C ǫ ln(1 /p )) µ . Case (2):
We have µ − i ≤ C ǫ ′ i ln(1 /p ) , and pI i [ f ] ≥ (1 − C ǫ ln(1 /p )) µ . N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 13
We remark that in Claim 3.3, it is necessary that C depend on η ; this is evi-denced e.g. by the function f = 1 B in Section 4, with t = 1 , s = 3 and i = 1 . Proof.
By Lemma 3.1 and equation (3.2), we have pI i [ f ] log p (cid:18) pµ + i µ (cid:19) = p ( µ + i − µ − i ) log p (cid:18) pµ + i µ (cid:19) ≤ F (cid:0) µ + i , µ − i (cid:1) − G (cid:0) µ + i , µ − i (cid:1) ≤ ǫ ′ i . We now split into two cases.Case (a): µ + i ≤ (1 − η ) µp .If µ + i ≤ (1 − η ) µp , then Case (1) of Claim 3.3 must occur, provided we take C to be sufficiently large. Indeed, we then have pI i [ f ] log p (1 − η ) = p ( µ + i − µ − i ) log p (1 − η ) ≤ p ( µ + i − µ − i ) log p (cid:18) pµ + i µ (cid:19) ≤ ǫ ′ i , which gives pI i [ f ] ≤ ( − η ) ǫ ′ i ln(1 /p ) ≤ C ǫ ′ i ln(1 /p ) , provided we choose C ≥ / (ln(2 / (2 − η ))) . This in turn implies that µ − i = µ − p ( µ + i − µ − i ) = µ − pI i [ f ] ≥ µ − C ǫ ′ i ln(1 /p ) ≥ µ − C ǫµ ln(1 /p ) , so Case (1) occurs, as asserted.Case (b): µ + i ≥ (cid:0) − η (cid:1) µp .If µ + i ≥ (1 − η ) µp , then Case (2) of Claim 3.3 must occur. Indeed, since µ − i ≤ µ ,we have pI i [ f ] = p ( µ + i − µ − i ) ≥ (cid:0) − η − p (cid:1) µ ≥ ηµ . We now have log p (cid:18) pµ + i µ (cid:19) ≤ ǫ ′ i p ( µ + i − µ − i ) ≤ ǫ ′ i ηµ ≤ ǫ ′ i ηpµ + i . Hence, pµ + i µ ≥ p ǫ ′ i / ( ηpµ + i ) = exp (cid:18) − ǫ ′ i ln(1 /p ) ηpµ + i (cid:19) . Using the fact that − e − x ≤ x for all x ≥ , we have (1 − p ) µ − i µ = 1 − pµ + i µ ≤ − exp (cid:18) − ǫ ′ i ln(1 /p ) ηpµ + i (cid:19) ≤ ǫ ′ i ln(1 /p ) ηpµ + i . This implies µ − i ≤ (cid:18) µηpµ + i (cid:19) (cid:18) − p (cid:19) ǫ ′ i ln(1 /p ) ≤ (cid:18) η (2 − η ) (cid:19) (cid:18) η (cid:19) ǫ ′ i ln(1 /p ) ≤ C ǫ ′ i ln(1 /p ) , provided we choose C ≥ η (2 − η ) . We now have pI i [ f ] = p ( µ + i − µ − i ) = µ − µ − i ≥ µ − C ǫ ′ i ln(1 /p ) ≥ µ − C ǫµ ln(1 /p ) , so Case (2) occurs, as asserted. (cid:3) We now prove a version of Claim 3.2 for small p and a general f (i.e., notnecessarily monotone increasing). Here, similarly to in the monotone case, weobtain that either µ − i is small, or else pI i [ f ] is small. Claim 3.4.
There exists an absolute constant C > such that if < p ≤ e − ,then for each i ∈ [ n ] , one of the following holds. Case (1):
We have pI i [ f ] ≤ C ǫ ′ i ln(1 /p ) , and µ − i ≥ (1 − C ǫ ln(1 /p )) µ . N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 14
Case (2):
We have µ − i ≤ C ǫ ′ i ln(1 /p ) , and pI i [ f ] ≥ (1 − C ǫ ln(1 /p )) µ .Proof. By (3.9), we have(3.10) pI i [ f ] − p (cid:12)(cid:12) µ + i − µ − i (cid:12)(cid:12) ≤ ǫ ′ i . Firstly, suppose that µ − i ≥ µ + i ; then µ − i ≥ µ , so clearly we have µ − i ≥ (1 − C ǫ ln(1 /p )) µ for any C > . Moreover, by Lemma 3.1 and (3.3), we have(3.11) (cid:0) µ − i − µ + i (cid:1) p ≤ H (cid:0) µ + i , µ − i (cid:1) − G (cid:0) µ + i , µ − i (cid:1) ≤ ǫ ′ i Combining (3.10) and (3.11) yields pI i [ f ] ≤ ǫ ′ i , so Case (1) holds.Secondly, suppose that µ + i > µ − i . By Lemma 3.1 and equation (3.2), we have p ( µ + i − µ − i ) log p (cid:18) pµ + i µ (cid:19) ≤ F (cid:0) µ + i , µ − i (cid:1) − G (cid:0) µ + i , µ − i (cid:1) ≤ ǫ ′ i . Similarly to in the proof of Claim 3.3, we now split into two cases.Case (a): µ + i ≤ µ p .If µ + i ≤ µ p , then Case (1) of Claim 3.3 must occur, provided we take C to besufficiently large. Indeed, we then have p ( µ + i − µ − i ) log p (1 / ≤ p ( µ + i − µ − i ) log p (cid:18) pµ + i µ (cid:19) ≤ ǫ ′ i , which, in combination with (3.10), gives pI i [ f ] ≤ ǫ ′ i ln(1 /p ) + ǫ ′ i ≤ C ǫ ′ i ln(1 /p ) ,provided we choose C ≥ / (ln 2) + 1 / . This in turn implies that µ − i = µ − p ( µ + i − µ − i ) ≥ µ − pI i [ f ] ≥ µ − C ǫ ′ i ln(1 /p ) ≥ µ − C ǫµ ln(1 /p ) , so Case (1) occurs, as asserted.Case (b): µ + i ≥ µ p .If µ + i ≥ µ p , then Case (2) of Claim 3.3 must occur. Indeed, since µ − i ≤ µ , wehave p ( µ + i − µ − i ) ≥ (cid:0) − p (cid:1) µ ≥ µ . We now have log p (cid:18) pµ + i µ (cid:19) ≤ ǫ ′ i p ( µ + i − µ − i ) ≤ ǫ ′ i µ ≤ ǫ ′ i pµ + i . Hence, pµ + i µ ≥ p ǫ ′ i / ( pµ + i ) = exp (cid:18) − ǫ ′ i ln(1 /p ) pµ + i (cid:19) . Using the fact that − e − x ≤ x for all x ≥ , we have (1 − p ) µ − i µ = 1 − pµ + i µ ≤ − exp (cid:18) − ǫ ′ i ln(1 /p ) pµ + i (cid:19) ≤ ǫ ′ i ln(1 /p ) pµ + i . This implies µ − i ≤ (cid:18) µpµ + i (cid:19) (cid:18) − p (cid:19) ǫ ′ i ln(1 /p ) ≤ ǫ ′ i ln(1 /p )1 − e − ≤ C ǫ ′ i ln(1 /p ) , provided we choose C ≥ − e − . We now have pI i [ f ] ≥ p ( µ + i − µ − i ) = µ − µ − i ≥ µ − C ǫ ′ i ln(1 /p ) ≥ µ − C ǫµ ln(1 /p ) , so Case (2) occurs, as asserted. (cid:3) N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 15
Either µ is fairly small, or very close to 1. Here, we show that thereexists a constant c > such that either µ = 1 − O (cid:18) ǫ ln(1 /p )log ( ǫ ln(1 /p ) ) (cid:19) (i.e., µ is veryclose to 1), or else µ < − c (i.e., µ is bounded away from 1). For a general f (and < p ≤ / , we obtain this by applying the p -biased isoperimetric inequality to thecomplement of f : ˜ f ( x ) = 1 − f ( x ) . For monotone f (and < p < ), we apply the p -biased isoperimetric inequality to the dual of f : f ∗ ( x ) = 1 − f ( x ) = 1 − f (1 − x ) . Claim 3.5.
Let < p ≤ / . Then we either have µ ≥ − C ǫ ln(1 /p )ln (cid:16) ǫ ln(1 /p ) (cid:17) , or else µ ≤ − c , where C , c > are absolute constants with c < / .Proof. Note that µ p ( ˜ f ) = 1 − µ p ( f ) and that I p [ ˜ f ] = I p [ f ] . By assumption, wehave pI [ f ] = µ (log p µ + ǫ ) . On the other hand, applying Theorem 1.6 to ˜ f , weobtain pI [ f ] = pI [ ˜ f ] ≥ (1 − µ ) log p (1 − µ ) . Combining these two facts, we obtain µ (cid:0) log p µ + ǫ (cid:1) ≥ (1 − µ ) log p (1 − µ ) . Suppose that δ := 1 − µ ≤ c , where c > is to be chosen later. Then δ log p ( δ ) ≤ (1 − δ ) (cid:0) log p (1 − δ ) + ǫ (cid:1) = (1 − δ ) ln (cid:16) − δ (cid:17) ln (cid:16) p (cid:17) + ǫ ≤ δ ln (cid:16) p (cid:17) + ǫ, where the last inequality holds provided c is sufficiently small. Hence, δ (cid:18) ln (cid:18) δ (cid:19) − (cid:19) ≤ ǫ ln (cid:18) p (cid:19) . Provided c is sufficiently small, this implies that δ = O ǫ ln(1 /p )ln (cid:16) ǫ ln(1 /p ) (cid:17) , proving the claim. (cid:3) Claim 3.6.
For any η > , there exist C = C ( η ) and c = c ( η ) > such thatthe following holds. Suppose that < p ≤ − η , and suppose that f is monotoneincreasing. Then we either have µ ≥ − C ǫ ln (cid:0) ǫ (cid:1) , or else µ ≤ − c .Proof. Note that f ∗ is monotone increasing, since f is. Moreover, µ − p ( f ∗ ) =1 − µ p ( f ) and I − p [ f ∗ ] = I p [ f ] . By assumption, we have pI p [ f ] = µ (log p µ + ǫ ) .On the other hand, applying Theorem 1.6 to f ∗ , we obtain (1 − p ) I p [ f ] = (1 − p ) I − p [ f ∗ ] ≥ (1 − µ ) log − p (1 − µ ) . N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 16
Combining these two facts, we obtain µ (cid:0) log p µ + ǫ (cid:1) ≥ p − p (1 − µ ) log − p (1 − µ ) . Suppose that δ := 1 − µ ≤ c , where c = c ( η ) > is to be chosen later. Then p − p δ log − p ( δ ) ≤ (1 − δ ) (cid:0) log p (1 − δ ) + ǫ (cid:1) = (1 − δ ) ln (cid:16) − δ (cid:17) ln (cid:16) p (cid:17) + ǫ ≤ δ ln (cid:16) p (cid:17) + ǫ, (3.12)where the last inequality holds provided c is sufficiently small. Observe that ln (cid:16) − p (cid:17) = Θ η ( p ) . Hence,(3.13) p − p δ log − p ( δ ) = p − p δ ln (cid:0) δ (cid:1) ln (cid:16) − p (cid:17) = Θ η p − p δ ln (cid:0) δ (cid:1) p ! = Θ η (cid:18) δ ln (cid:18) δ (cid:19)(cid:19) . Combining (3.12) and (3.13), we obtain Θ η ( δ ln(1 /δ )) − δη ≤ Θ η ( δ ln(1 /δ )) − δ ln (1 / (1 − η )) ≤ Θ η ( δ ln(1 /δ )) − δ ln (1 /p ) ≤ ǫ, using the fact that − η ≤ e − η for all η ∈ R . This in turn implies that δ = O η ǫ ln (cid:0) ǫ (cid:1) ! provided c is sufficiently small depending on η , proving the claim. (cid:3) There exists an influential coordinate.
We now show that unless µ is veryclose to , there must exist a coordinate whose influence is large. This coordinatewill be used in the inductive step of the proof of our two stability theorems. First,we deal with the case of small p and general f (i.e., f not necessarily monotoneincreasing). Claim 3.7.
For any ζ ∈ (0 , c / , the following holds provided c is sufficientlysmall (depending on the absolute constants C , C and c ). Suppose that < p < ζ and ǫ ln(1 /p ) ≤ c . If µ < − C ǫ ln(1 /p )ln ( ǫ ln(1 /p ) ) , then there exists i ∈ [ n ] for which Case (2)of Claim 3.4 occurs, i.e. µ − i ≤ C ǫ ′ i ln(1 /p ) and pI i [ f ] ≥ (1 − C ǫ ln(1 /p )) µ . (Here, C is the absolute constant from Claim 3.4, and C , c are the absolute constantsfrom Claim 3.5.)Proof. We prove the claim by induction on n .If n = 1 and µ < − C ǫ ln(1 /p )ln ( ǫ ln(1 /p ) ) , then by Claim 3.5, we have µ < − c , andtherefore f ≡ or f = 1 { x =1 } . (If f = 1 { x =0 } then µ = 1 − p > − ζ > − c / .)Hence, we have µ − = 0 , so Case (2) must occur for the coordinate 1, verifying thebase case.We now do the inductive step. Let n ≥ , and assume the claim holds when n is replaced by n − . Let f be as in the statement of the claim; then by Claim 3.5,we have µ ≤ − c . Suppose for a contradiction that f has Case (1) of Claim 3.4 N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 17 occurring for each i ∈ [ n ] . First, suppose that ǫ − i ≥ ǫ for each i ∈ [ n ] . Fix any i ∈ [ n ] . By (3.2), we have ≤ ǫ ′ i ≤ p ( ǫ − ǫ + i ) µ + i , so ǫ + i ≤ ǫ and therefore(3.14) I i [ f ] ≤ p C ǫ ′ i ln(1 /p ) ≤ C (cid:0) ǫ − ǫ + i (cid:1) µ + i ln(1 /p ) ≤ C c µ + i , using our assumption that ǫ ln(1 /p ) ≤ c . Hence, µ + i − µ ≤ | µ + i − µ − i | ≤ I i [ f ] ≤ C c µ + i , so µ + i ≤ µ − C c ≤ − c − C c < − C c ln(1 /c ) ≤ − C ǫ ln(1 /p )ln (cid:16) ǫ ln(1 /p ) (cid:17) ≤ − C ǫ + i ln(1 /p )ln (cid:16) ǫ + i ln(1 /p ) (cid:17) , provided c is sufficiently small (depending on C , C and c ). It follows that f i → satisfies the hypothesis of the claim, for each i ∈ [ n ] . Hence, by the inductionhypothesis, there exists j ∈ [ n ] \ { i } such that f i → has Case (2) of Claim 3.4occurring for the coordinate j , so pI j [ f i → ] ≥ (cid:0) − C ǫ + i ln(1 /p ) (cid:1) µ + i . We now have I j [ f ] ≥ pI j [ f i → ] ≥ (cid:0) − C ǫ + i ln(1 /p ) (cid:1) µ + i ≥ (1 − C ǫ ln(1 /p )) µ + i ≥ (1 − C c ) µ + i , but this contradicts the fact that (3.14) holds when i is replaced by j , provided c is sufficiently small (depending on C ).We may assume henceforth that there exists i ∈ [ n ] such that ǫ − i < ǫ . Fix sucha coordinate i . Since Case (1) occurs for the coordinate i , we have(3.15) µ − i ≥ (1 − C ǫ ln(1 /p )) µ ≥ (1 − C c ) µ. On the other hand, we have µ − i ≤ µ − p ≤ − c − ζ < − c − c / ≤ − C c ln(1 /c ) ≤ − C ǫ ln(1 /p )ln (cid:16) ǫ ln(1 /p ) (cid:17) < − C ǫ − i ln(1 /p )ln (cid:16) ǫ − i ln(1 /p ) (cid:17) , provided c is sufficiently small (depending on C and c ). Hence, f i → satisfiesthe hypotheses of the claim. Therefore, by the induction hypothesis, there exists j ∈ [ n ] \ { i } such that f i → has Case (2) of Claim 3.4 occurring for the coordinate j , so pI j [ f i → ] ≥ (cid:0) − C ǫ − i ln(1 /p ) (cid:1) µ − i . Therefore, we have pI j [ f ] ≥ p (1 − p ) I j [ f i → ] ≥ (1 − p ) (cid:0) − C ǫ − i ln(1 /p ) (cid:1) µ − i > (1 − p ) (1 − C ǫ ln(1 /p )) (1 − C c ) µ ≥ (1 − C c ) µ, using (3.15) for the third inequality, contradicting the fact that f satisfies Case (1)of Claim 3.4 for the coordinate j , provided c is sufficiently small (depending on C ). This completes the inductive step, proving the claim. (cid:3) Now we deal with the case of p bounded away from and bounded from aboveby / , and arbitrary f . N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 18
Claim 3.8.
For each ζ > , the following holds provided c is sufficiently smalldepending on ζ . Suppose that ζ < p ≤ / and ǫ ln(1 /p ) ≤ c . If µ < − C ǫ ln(1 /p )ln ( ǫ ln(1 /p ) ) ,then there exists i ∈ [ n ] for which Case (2) of Claim 3.2 occurs, i.e. min { µ − i , µ + i } ≤ C ǫ ′ i and I i [ f ] ≥ (1 − C ǫ ) µ . (Here, C = C ( ζ ) is the constant from Claim 3.2,and C is the absolute constant from Claim 3.5.)Proof. If n = 1 and µ < − C ǫ ln(1 /p )ln ( ǫ ln(1 /p ) ) , then we must have either f ≡ , f = 1 { x =1 } or f = 1 { x =0 } . Hence, we have min { µ + i , µ − } = 0 , so Case (2) of Claim 3.2 mustoccur for the coordinate 1, verifying the base case.We now do the inductive step. Let n ≥ , and assume the claim holds when n is replaced by n − . Let f be as in the statement of the claim; then by Claim3.5, we have µ ≤ − c . Suppose for a contradiction that f has Case (1) ofClaim 3.2 occurring for each i ∈ [ n ] . First, suppose that ǫ − i ≥ ǫ for each i ∈ [ n ] .Then almost exactly the same argument as in the proof of Claim 3.7 yields acontradiction, provided c is sufficiently small depending on ζ . Therefore, we mayassume henceforth that there exists i ∈ [ n ] such that ǫ − i < ǫ . By assumption, Case1 of Claim 3.2 occurs for the coordinate i , and therefore min { µ + i , µ − i } ≥ (1 − C ǫ ) µ .It follows that µ − i = µ − pµ + i − p ≤ − p (1 − C ǫ )1 − p µ = µ + pC ǫµ − p ≤ − c + C ǫ < − C c ln(1 /c ) ≤ − C ǫ ln(1 /p )ln (cid:16) ǫ ln(1 /p ) (cid:17) < − C ǫ − i ln(1 /p )ln (cid:16) ǫ − i ln(1 /p ) (cid:17) , provided c is sufficiently small depending on ζ . Hence, f i → satisfies the hypothesesof the claim. Therefore, by the induction hypothesis, there exists j ∈ [ n ] \ { i } suchthat f i → has Case (2) of Claim 3.4 occurring for the coordinate j , so I j [ f i → ] ≥ (cid:0) − C ǫ − i (cid:1) µ − i . We now have I j [ f ] ≥ (1 − p ) I j [ f i → ] ≥ (1 − p ) (cid:0) − C ǫ − i (cid:1) µ − i > (1 − p ) (1 − C ǫ ) µ ≥ (1 − C c / ln(2)) µ, contradicting the fact that f satisfies Case (1) of Claim 3.2 for the coordinate j ,provided c is sufficiently small depending on C (i.e., on ζ ). This completes theinductive step, proving the claim. (cid:3) Finally, we deal with the case of monotone f and all p bounded away from 1. Claim 3.9.
For each η > , the following holds provided c is sufficiently smalldepending on η . Let < p ≤ − η , and suppose f is monotone increasing. If µ < − C ǫ ln(1 /p )ln ( ǫ ln(1 /p ) ) , then there exists i ∈ [ n ] for which Case (2) of Claim 3.3 occurs.(Here, C = C ( η ) is the constant from Claim 3.6.)Proof. We prove the claim by induction on n .If n = 1 and µ < − C ǫ ln(1 /p )ln ( ǫ ln(1 /p ) ) , then since µ < we must have either f ≡ or f = 1 { x =1 } , so µ − = 0 . Hence, Case (2) of Claim 3.3 occurs for the coordinate 1,verifying the base case. N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 19
We now do the inductive step. Let n ≥ , and assume the claim holds when n is replaced by n − . Let f be as in the statement of the claim; then by Claim 3.6,we have µ ≤ − c . Suppose for a contradiction that f has Case (1) of Claim 3.3occurring for each i ∈ [ n ] . First, suppose that ǫ − i ≥ ǫ for each i ∈ [ n ] . Fix any i ∈ [ n ] . Then almost exactly the same argument as in the proof of Claim 3.7 (usingClaim 3.6 in place of Claim 3.5) yields a contradiction.We may therefore assume henceforth that there exists i ∈ [ n ] such that ǫ − i < ǫ .Since Case (1) of Claim 3.3 occurs for the coordinate i , we have µ − i ≥ (1 − C ǫ ln(1 /p )) µ ≥ (1 − C c ) µ. On the other hand, we have µ − i ≤ µ < − C ǫ ln(1 /p )ln (cid:16) ǫ ln(1 /p ) (cid:17) < − C ǫ − i ln(1 /p )ln (cid:16) ǫ − i ln(1 /p ) (cid:17) , so f i → satisfies the hypotheses of the claim. Hence, by the induction hypothesis,there exists j ∈ [ n ] \ { i } such that f i → has Case (2) of Claim 3.3 occurring for thecoordinate j , so pI j [ f i → ] ≥ (cid:0) − C ǫ − i ln(1 /p ) (cid:1) µ − i . We now have pI j [ f ] ≥ p (1 − p ) I j [ f i → ] ≥ (1 − p ) (cid:0) − C ǫ − i ln(1 /p ) (cid:1) µ − i > (1 − p ) (1 − C ǫ ln(1 /p )) (1 − C c ) µ ≥ η (1 − C c ) µ, contradicting the fact that f satisfies Case (1) of Claim 3.3 for the coordinate j ,provided c is sufficiently small depending on η . This completes the inductive step,proving the claim. (cid:3) Bootstrapping.
Our final required ingredient is a ‘bootstrapping’ argument,which says that if min (cid:8) µ − i , µ + i (cid:9) is ‘somewhat’ small, then it must be ‘very’ small. Claim 3.10.
Let ζ ∈ (0 , / . There exist C = C ( ζ ) > and c = c ( ζ ) > such that the following holds. Let ζ < p ≤ . If µ − i ≤ c µ , then µ − i ≤ C (cid:0) ǫ − ǫ + i (cid:1) ln(1 /p )ln (cid:0) / (cid:0) ( ǫ − ǫ + i ) ln(1 /p ) (cid:1)(cid:1) µ, and if µ + i ≤ c µ , then µ + i ≤ C (cid:0) ǫ − ǫ − i (cid:1) ln(1 /p )ln (cid:0) / (cid:0) ( ǫ − ǫ − i ) ln(1 /p ) (cid:1)(cid:1) µ. Proof.
Let c = c ( ζ ) > to be chosen later. First suppose that µ − i ≤ µ + i , andwrite δ := µ − i /µ ; then δ ≤ c . Using (3.2), we have (1 − p ) µ − i log p (cid:0) µ − i (cid:1) + pµ + i log p µ + i − µ log p µ + pI i [ f ]= (cid:0) ǫ − ǫ + i (cid:1) pµ + i + ( ǫ − ǫ − i ) (1 − p ) µ − i ≤ (cid:0) ǫ − ǫ + i (cid:1) µ + ǫ (1 − p ) µ − i , (3.16) N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 20 the last inequality following from the fact that pµ + i = µ − (1 − p ) µ − i ≤ µ . Observethat LHS = (1 − p ) µ − i log p (cid:0) µ − i (cid:1) + pµ + i log p (cid:0) pµ + i (cid:1) − µ log p µ + pI i [ f ] − pµ + i ≥ (1 − p ) µ − i log p (cid:0) µ − i (cid:1) + pµ + i log p ( µ ) − µ log p ( µ ) − pµ − i = (1 − p ) µ − i log p (cid:18) µ − i µ (cid:19) − pµ − i . (3.17)Combining (3.16) and (3.17) and rearranging, we obtain(3.18) (cid:18) µ − i µ (cid:19) (cid:18) log p (cid:18) µ − i µ (cid:19) − p − p − ǫ (cid:19) ≤ ǫ − ǫ + i − p ≤ ǫ − ǫ + i ) . It follows that δ (cid:18) ln(1 /δ ) − p ln(1 /p )1 − p − ǫ ln(1 /p ) (cid:19) ≤ ǫ − ǫ + i ) ln(1 /p ) , and therefore δ (cid:18) ln(1 /δ ) − e − c (cid:19) ≤ ǫ − ǫ + i ) ln(1 /p ) , using the fact that p ln(1 /p ) / (1 − p ) ≤ /e whenever ≤ p ≤ / . Since δ ≤ c , if c is sufficiently small this clearly implies that δ = O ζ ( ǫ − ǫ + i ) ln(1 /p )ln (cid:16) ǫ − ǫ + i ) ln(1 /p ) (cid:17) , as required.Now suppose that µ + i ≤ µ − i , and write δ := µ + i /µ ; then δ ≤ c . Using (3.2), wehave (1 − p ) µ − i log p (cid:0) µ − i (cid:1) + pµ + i log p µ + i − µ log p µ + pI i [ f ]= (cid:0) ǫ − ǫ + i (cid:1) pµ + i + ( ǫ − ǫ − i ) (1 − p ) µ − i ≤ ǫpµ + i + ( ǫ − ǫ − i ) µ. (3.19)Observe thatLHS = (1 − p ) µ − i log p (cid:0) (1 − p ) µ − i (cid:1) + pµ + i log p (cid:0) µ + i (cid:1) − µ log p ( µ ) + pI i [ f ] − (1 − p ) µ − i log p (1 − p ) ≥ (1 − p ) µ − i log p ( µ ) + pµ + i log p (cid:0) µ + i (cid:1) − µ log p ( µ ) + p ( µ − i − µ + i ) − (1 − p ) µ − i log p (1 − p )= pµ + i log p (cid:18) µ + i µ (cid:19) + µ − i ( p − (1 − p ) log p (1 − p )) − pµ + i = pµ + i log p (cid:18) µ + i µ (cid:19) + K ( p ) µ − i − pµ + i ≥ pµ + i log p (cid:18) µ + i µ (cid:19) − pµ + i . (3.20)Combining (3.19) and (3.20) and rearranging, we obtain(3.21) (cid:18) µ + i µ (cid:19) (cid:18) log p (cid:18) µ + i µ (cid:19) − − ǫ (cid:19) ≤ ǫ − ǫ − i p ≤ ζ ( ǫ − ǫ − i ) . N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 21
It follows that δ (ln(1 /δ ) − ln(1 /p ) − ǫ ln(1 /p )) ≤ ζ ( ǫ − ǫ − i ) ln(1 /p ) , and therefore δ (ln(1 /δ ) − ln(1 /ζ ) − c ) ≤ ζ ( ǫ − ǫ − i ) ln(1 /p ) . Since δ ≤ c , if c is sufficiently small (depending on ζ ), this clearly implies that δ = O ζ ( ǫ − ǫ − i ) ln(1 /p )ln (cid:16) ǫ − ǫ − i ) ln(1 /p ) (cid:17) , as required. (cid:3) We now prove a bootstrapping claim suitable for use in the cases where p ≤ ζ and f is arbitrary, or where p ≤ − η and f is monotone increasing. Claim 3.11.
Let η > . There exist C = C ( η ) > and c = c ( η ) > such thatif p ≤ − η and µ − i ≤ c µ , then µ − i ≤ C (cid:0) ǫ − ǫ + i (cid:1) ln(1 /p )ln (cid:0) / (cid:0)(cid:0) ǫ − ǫ + i (cid:1) ln(1 /p ) (cid:1)(cid:1) µ. Proof.
As in the proof of Claim 3.10, we have(3.22) (cid:18) µ − i µ (cid:19) (cid:18) log p (cid:18) µ − i µ (cid:19) − p − p − ǫ (cid:19) ≤ ǫ − ǫ + i − p ≤ ǫ − ǫ + i η . Writing δ := µ − i µ ≤ c , we obtain δ (cid:18) ln(1 /δ ) − eη − c (cid:19) ≤ δ (cid:18) ln(1 /δ ) − p ln(1 /p )1 − p − ǫ ln(1 /p ) (cid:19) ≤ ln(1 /p ) O η (cid:0) ǫ − ǫ + i (cid:1) , using the fact that p ln(1 /p ) / (1 − p ) ≤ / ( eη ) whenever ≤ p ≤ − η . Provided c = c ( η ) > is sufficiently small, this implies that δ = O η (cid:0) ǫ − ǫ + i (cid:1) ln(1 /p )ln (cid:0) / (cid:0)(cid:0) ǫ − ǫ + i (cid:1) ln(1 /p ) (cid:1)(cid:1) ! , as required. (cid:3) Inductive proofs of Theorems 1.7 and 1.8.
Proof of Theorem 1.7.
First, we choose any ζ ∈ (0 , c / (where c is the absoluteconstant from Claim 3.5), and we deal with the case of p < ζ , using Claim 3.7. Inthis case, we prove that the conclusion of Theorem 1.7 holds with S a monotoneincreasing subcube.We proceed by induction on n . If n = 1 , then f is the indicator functionof a monotone increasing subcube unless f = 1 { x =0 } , so we may assume that f = 1 { x =0 } . Then µ p ( f ) = 1 − p > − ζ > − c , so by Claim 3.5, we have µ p ( f ) ≥ − C ǫ ln(1 /p )ln (cid:16) ǫ ln(1 /p ) (cid:17) , so the conclusion of the theorem holds with S = { , } .We now do the inductive step. Let n ≥ , and assume that Theorem 1.7 holdswhen n is replaced by n − . Let f : { , } n → { , } satisfy the hypotheses of N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 22
Theorem 1.7. We may assume throughout that µ p ( f ) ≤ − c , otherwise by Claim3.5, we have µ p ( f ) ≥ − C ǫ ln(1 /p )ln (cid:16) ǫ ln(1 /p ) (cid:17) , so the conclusion of the theorem holds with S = { , } n . Since µ p ( f ) ≤ − c , byClaim 3.7, there exists i ∈ [ n ] such that µ − i ≤ C ǫµ ln(1 /p ) , so if c is a sufficientlysmall absolute constant, we have µ − i ≤ c µ , where c is the absolute constant weobtain by applying Claim 3.11 with η = 1 − ζ . Hence, µ − i satisfies the hypothesisof Claim 3.11. Therefore, we have(3.23) µ − i ≤ C (cid:0) ǫ − ǫ + i (cid:1) ln(1 /p )ln (cid:0) / (cid:0) ( ǫ − ǫ + i ) ln(1 /p ) (cid:1)(cid:1) µ, where C is the absolute constant we obtain by applying Claim 3.11 with η = 1 − ζ .In particular, we have ǫ + i ≤ ǫ . By applying the induction hypothesis to f i → , weobtain µ p ( f i → ∆1 S T ) ≤ C ǫ + i ln(1 /p ) µ + i ln (cid:16) ǫ + i ln(1 /p ) (cid:17) for some monotone increasing subcube S T = { x ∈ { , } [ n ] \{ i } : x j = 1 ∀ j ∈ T } ,where T ⊂ [ n ] . Therefore, writing S T ∪{ i } := { x ∈ { , } n : x j = 1 ∀ j ∈ T ∪ { i }} , we have µ p (cid:0) f ∆1 S T ∪{ i } (cid:1) ≤ (1 − p ) µ − i + pµ p ( f i → ∆1 S T ) ≤ (1 − p ) C ( ǫ − ǫ + i ) ln(1 /p ) µ ln (cid:18) [ ǫ − ǫ + i ] ln(1 /p ) (cid:19) + C ǫ + i ln(1 /p ) pµ + i ln (cid:16) ǫ + i ln(1 /p ) (cid:17) ≤ (cid:0) C (cid:0) ǫ − ǫ + i (cid:1) + C ǫ + i (cid:1) µ ln(1 /p )ln (cid:16) ǫ ln(1 /p ) (cid:17) ≤ C ǫµ ln(1 /p )ln (cid:16) ǫ ln(1 /p ) (cid:17) , provided C ≥ C , using (3.23). Hence, the conclusion of the theorem holds with S = S T ∪{ i } . This completes the inductive step, proving the theorem in the case p < ζ .Now we prove the theorem in the case ζ ≤ p ≤ / .We proceed again by induction on n . If n = 1 , then as before the theorem holdstrivially. Let n ≥ , and assume Theorem 1.7 holds when n is replaced by n − .Let f : { , } n → { , } satisfy the hypotheses of Theorem 1.7. As before, we mayassume throughout that µ p ( f ) ≤ − c , otherwise by Claim 3.5, we have µ p ( f ) ≥ − C ǫ ln(1 /p )ln (cid:16) ǫ ln(1 /p ) (cid:17) , so the conclusion of the theorem holds with S = { , } n . Since µ p ( f ) ≤ − c , byClaim 3.8 (applied with ζ = ζ ), provided c is sufficiently small depending on ζ ,there exists i ∈ [ n ] such that min { µ − i , µ + i } ≤ C ( ζ ) ǫ ′ i ≤ C ( ζ ) ǫµ , so we have min { µ − i , µ + i } ≤ C ( ζ ) ǫµ ≤ C ( ζ ) c / ln(1 /p ) ≤ C ( η ) c / ln(2) ≤ c ( ζ ) µ, N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 23 provided c ≤ ( c ( ζ ) ln 2) /C ( ζ ) . Hence, either µ − i or µ + i satisfies the hypothesisof Claim 3.10 (with ζ = ζ ). Suppose that µ − i ≤ c ( ζ ) µ (the other case is verysimilar). Then, by Claim 3.10, we have(3.24) µ − i ≤ C ( ζ )( ǫ − ǫ + i ) µ ln(1 /p )ln (cid:16) ǫ − ǫ + i ) ln(1 /p ) (cid:17) , and so in particular, ǫ + i ≤ ǫ . By applying the induction hypothesis to f i → , weobtain µ p ( f i → ∆1 S ′ ) ≤ C ǫ + i ln(1 /p ) µ + i ln (cid:16) ǫ + i ln(1 /p ) (cid:17) for some subcube S ′ = { x ∈ { , } [ n ] \{ i } : x j = a j ∀ j ∈ T } , where T ⊂ [ n ] and a j ∈ { , } for each j ∈ T . Therefore, writing S := { x ∈ { , } n : x j = a j ∀ j ∈ T, x i = 1 } , we have µ p ( f ∆1 S ) ≤ (1 − p ) µ − i + pµ p ( f i → ∆1 S ′ ) ≤ (1 − p ) C ( ζ )( ǫ − ǫ + i ) µ ln(1 /p )ln (cid:16) ǫ − ǫ + i ) ln(1 /p ) (cid:17) + C ǫ + i ln(1 /p ) pµ + i ln (cid:16) ǫ + i ln(1 /p ) (cid:17) ≤ (cid:0) C ( ζ ) (cid:0) ǫ − ǫ + i (cid:1) + C ǫ + i (cid:1) µ ln(1 /p )ln (cid:16) ǫ ln(1 /p ) (cid:17) ≤ C ǫµ ln(1 /p )ln (cid:16) ǫ ln(1 /p ) (cid:17) , provided C ≥ C ( ζ ) , using (3.24). This completes the inductive step, proving thetheorem in the case ζ ≤ p ≤ / , and completing the proof of Theorem 1.7.The inductive proof of Theorem 1.8 is very similar indeed, except that the con-stants are allowed to depend upon η (where η is as in the statement of Theorem1.8); we omit the details. (cid:3) Sharpness of Theorems 1.7 and 1.8
Theorem 1.7 is best possible up to the values of the absolute constants c and C . This can be seen by taking f = 1 A , where A = { x ∈ { , } n : x i = 1 ∀ i ∈ [ t ] }∪ { x ∈ { , } n : x i = 1 ∀ i ∈ [ t + s ] \ { t } , x t = 0 }\ { x ∈ { , } n : x i = 1 ∀ i ∈ [ t + s ] \ { t + 1 } , x t +1 = 0 } , for s, t ∈ N with s ≥ . Let < p ≤ / . We have µ p ( A ) = p t , and I i [ A ] = p t − if ≤ i ≤ t − − p s − ) p t − if i = t ; p t + s − if i = t + 1;2(1 − p ) p t + s − if t + 2 ≤ i ≤ t + s ;0 if i > t + s. Hence,(4.1) I p [ A ] = p t − (cid:0) t + 2( s − − p ) p s − (cid:1) . N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 24
On the other hand, it is easy to see that(4.2) µ p ( A ∆ S ) µ p ( A ) = µ p ( A ∆ S ) p t ≥ − p ) p s − for all subcubes S , with equality if and only if S = { x ∈ { , } n : x i = 1 ∀ i ∈ [ t ] } := C . Indeed, note that µ p ( C \ A ) = µ p ( A \ C ) = (1 − p ) p t + s − . Suppose that S = { x ∈ { , } n : x i = a i ∀ i ∈ F } , where F ⊂ [ n ] and a i ∈ { , } for all i ∈ F . Ifthere exists i ∈ F ∩ [ t ] such that a i = 0 , then S ∩ C = ∅ and therefore µ p ( A ∆ S ) ≥ µ p ( A \ S ) ≥ µ p ( A ∩ C ) = p t − (1 − p ) p t + s − > − p ) p t + s − , the last inequality using the fact that s ≥ and p (1 − p ) ≤ / . If [ t ] \ F = ∅ ,say j ∈ [ t ] \ F , then for any x ∈ S ∩ C , we have x − e j ∈ S \ C , and therefore µ p ( S \ C ) ≥ − pp µ p ( S ∩ C ) ≥ µ p ( S ∩ C ) . Hence,(4.3) µ p ( S \ A ) ≥ µ p ( S \ C ) − µ p ( A \ C ) ≥ µ p ( S ∩ C ) − (1 − p ) p t + s − . On the other hand, we have(4.4) µ p ( A \ S ) ≥ µ p ( A ∩ C ) − µ p ( S ∩ C ) = p t − (1 − p ) p t + s − − µ p ( S ∩ C ) . Summing the inequalities (4.3) and (4.4), we obtain µ p ( A ∆ S ) ≥ p t − − p ) p t + s − > − p ) p t + s − , the last inequality using the fact that s ≥ and p (1 − p ) ≤ / . Hence, we mayassume that [ t ] ⊂ F and that a i = 1 for all i ∈ [ t ] , so in particular S ⊂ C . Supposethat F \ [ t ] = ∅ . Then µ p ( S ) ≤ (1 − p ) µ p ( C ) = (1 − p ) p t , and therefore µ p ( A \ S ) ≥ µ p (( A ∩ C ) \ S ) + µ p ( A \ C ) ≥ µ p ( A ∩ C ) − µ p ( S ) + µ p ( A \ C )= p t − (1 − p ) p t + s − − µ p ( S ) + (1 − p ) p t + s − = p t − µ p ( S ) ≥ p t − (1 − p ) p t = p t +1 > − p ) p t + s − , the last inequality using the fact that s ≥ and p (1 − p ) ≤ / . The only remainingcase is S = C , where equality holds in (4.2).It follows from (4.1) and (4.2) that if ǫ := 2( s − − p ) p s − , then pI p [ A ] = µ p ( A )(log p ( µ p ( A )) + ǫ ) , but µ p ( A ∆ S ) µ p ( A ) ≥ ǫs − δ, for all subcubes S . We have ( s − p s − ≥ ǫ , so writing s − x/ ln(1 /p ) , we get xe − x ≥ ǫ ln(1 /p ) , which implies x ≤ (cid:18) ǫ ln(1 /p ) (cid:19) , N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 25 or equivalently, s − ≤ /p ) ln (cid:18) ǫ ln(1 /p ) (cid:19) . Hence, δ ≥ ǫ ln(1 /p )2 ln (cid:16) ǫ ln(1 /p ) (cid:17) , showing that Theorem 1.7 is best possible up to the value of C . Moreover, weclearly require ǫ ln(1 /p ) < for the right-hand side of (1.5) to be non-negative, soin the statement of Theorem 1.7, it is necessary that c < .Observe that the above family A is not monotone increasing. To prove sharpnessfor Theorem 1.8, we may take f = 1 B , where B = { x ∈ { , } n : x i = 1 ∀ i ∈ [ t ] }∪{ x ∈ { , } n : x i = 1 ∀ i ∈ [ t + s ] \{ t } , x t = 0 } . for s, t ∈ N with s ≥ . Let < p < . We have µ p ( B ) = p t (1 + (1 − p ) p s − ) , and I pi [ B ] = p t − + (1 − p ) p t + s − if ≤ i ≤ t − − p s ) p t − if i = t ;(1 − p ) p t + s − if t + 1 ≤ i ≤ t + s ;0 if i > t + s. Hence, I p [ B ] = p t − ( t + (( t + s )(1 − p ) − p s − ) , and we have pI p [ B ] − µ p ( B ) log p ( µ p ( B )) µ p ( B ) ≤ ( s − − p ) p s − =: ǫ. On the other hand, we have µ p ( B ∆ S ) µ p ( B ) = µ p ( B ∆ S ) p t (1 + (1 − p ) p s − ) ≥ (1 − p ) p t + s − p t (1 + (1 − p ) p s − ) ≥ (1 − p ) p s − := δ for all subcubes S , with equality if and only if S = { x ∈ { , } n : x i = 1 ∀ i ∈ [ t ] } ,by a very similar argument to that above (for A ). Similarly to before, we obtain δ ≥ ǫ ln(1 /p )4 ln (cid:16) − pǫ ln(1 /p ) (cid:17) . Provided /e < p < , choosing s = ⌈ / ln(1 /p ) ⌉ + 1 yields δ ≥ ǫ ln(1 /p )4 ln (cid:16) − pǫ ln(1 /p ) (cid:17) = Ω(ln(1 / (1 − p ))) ǫ ln(1 /p )ln (cid:16) ǫ ln(1 /p ) (cid:17) ; in this case, writing p = 1 − η , we have ǫ = Θ(1 − p ) = O ( η ) = O ( η ) / ln(1 /p ) .This shows that Theorem 1.8 is best possible up to a constant factor depending on η , and that the statement of Theorem 1.8 holds only if c ( η ) = O ( η ) or C ( η ) =Ω(ln(1 /η )) , so the dependence on η cannot be removed.We note that B also demonstrates the sharpness of Theorem 1.7, but does nothave the nice property of log p ( µ p ( B )) ∈ N , so we think it worthwhile to includeboth examples. N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 26 Isoperimetry via Kruskal-Katona – Proof of Theorem 1.9, and anew proof of the ‘full’ edge isoperimetric inequality
In this section, we use the Kruskal-Katona theorem, the Margulis-Russo lemmaand some analytic and combinatorial arguments to prove Theorem 1.9, our biasedversion of the ‘full’ edge isoperimetric inequality, for monotone increasing sets. Wethen give the (very short) deduction of Theorem 1.1 (the ‘full’ edge isoperimetricinequality) from the p = 1 / case of Theorem 1.9, hence providing a new proof ofthe former — one that relies upon the Kruskal-Katona theorem. The Margulios-Russo Lemma.
We first recall the useful lemma of Margulis [32]and Russo [33].
Lemma 5.1 (Margulis, Russo) . Let
F ⊂ P ([ n ]) be a monotone increasing familyand let < p < . Then ddp µ p ( F ) (cid:12)(cid:12)(cid:12)(cid:12) p = p = I p [ F ] . . Lexicographic families in the Cantor space P ( N ) . We now give a formal defi-nition of the lexicographic families L λ (described less formally in the Introduction),and analyse some of their properties.We define L = ∅ and L = P ( N ) . For any λ ∈ (0 , , let the binary expansionof λ be(5.1) ∞ X j =1 − i j = λ where ≤ i < i < . . . (if the binary expansion is infinite), or(5.2) N X j =1 − i j = λ where ≤ i < i < . . . < i N (if the binary expansion is finite), and define L λ = [ j { S ⊂ N : S ∩ [ i j ] = [ i j ] \ { i k : k < j }} ⊂ P ( N ) . Equivalently, let T = { i , i , . . . } be the set whose characteristic vector correspondsto the binary expansion of λ , and let L λ = { S ⊂ [ n ] : S ≥ N \ T } be the initialsegment of the lexicographic ordering on P ( N ) ending at N \ T .Note that if the binary expansion of λ is finite, i.e. n λ ∈ N ∪{ } for some n ∈ N ,then L λ = L × P ( N \ [ n ]) , where L ⊂ P ([ n ]) is the lexicographic family of size n λ .We identify P ( N ) with the Cantor space { , } N , in the natural way. We let Σ bethe σ -algebra on P ( N ) generated by ∪ n ∈ N P ([ n ]) . By the countable unions propertyof σ -algebras, it is clear that L λ ∈ Σ for any λ ∈ [0 , .By the Kolmogorov Extension theorem (see [28], or e.g. [36] for a more modernexposition), there exists a unique probability measure µ ( N ) p on ( { , } N , Σ) such that µ ( N ) p ( A × A × . . . × A n × { , } × { , } × . . . ) = µ ( n ) p ( A × A × . . . × A n ) for all n ∈ N and all A , . . . , A n ⊂ { , } . We may call this measure the p -biasedproduct measure on { , } N . N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 27
Abusing notation slightly, we write µ p = µ ( N ) p when the underlying space { , } N is understood.If f : { , } N → { , } is Σ -measurable, we define influence of the i th coordinateon f by I pi [ f ] := Pr x ∼ µ p [ f ( x ) = f ( x ⊕ e i )] and we define the total influence of f by I p [ f ] := ∞ X i =1 I pi [ f ] . We remark that there exist Σ -measurable functions f : { , } N → { , } suchthat I p [ f ] = ∞ . However, the families L λ are better behaved, as we will shortlysee.Clearly, by the countable additivity of µ p , we have(5.3) µ p ( L λ ) = X j p i j − j +1 (1 − p ) j − , where the ( i j ) define the binary expansion of λ , as in (5.1) or (5.2).It is helpful to analyse the families L λ using the families ( L ⌊ λ n ⌋ / n ) n ∈ N , whichdepend upon only finitely many coordinates. To this end, for each λ ∈ [0 , andeach n ∈ N , we define L λ ( n ) := L ⌊ λ n ⌋ / n . For brevity, if p ∈ (0 , is fixed, wewrite r = r ( p ) := max { p, − p } , and if λ ∈ [0 , is fixed, we write L := L λ and L ( n ) := L λ ( n ) = L ⌊ λ n ⌋ n for each n ∈ N . Observe that for any λ ∈ [0 , , we have L ( n ) ⊂ L ( n + 1) ⊂ L for all n ∈ N . Claim 5.2.
Let < p < and let ≤ λ ≤ . Then µ p ( L \ L ( n )) ≤ r n +1 − r . Proof.
We may assume that < λ < . Let the binary expansion of λ be λ = X j − i j , where ≤ i < i < . . . , so that by definition, L = L λ = [ j { S ⊂ N : S ∩ [ i j ] = [ i j ] \ { i k : k < j }} ⊂ P ( N ) . Observe that for each n ∈ N , we have L ( n ) = [ j : i j ≤ n { S ⊂ N : S ∩ [ i j ] = [ i j ] \ { i k : k < j }} . For brevity, write C j := { S ⊂ N : S ∩ [ i j ] = [ i j ] \ { i k : k < j }} for each j ; then C j is a subcube whose set of fixed coordinates is [ i j ] , for each j , and we have L = [ j C j , L ( n ) = [ j : i j ≤ n C j . Hence, µ p ( L \ L ( n )) = X j : i j >n µ p ( C j ) ≤ r n +1 + r n +2 + . . . ≤ r n +1 − r , since the subcube C j has i j fixed coordinates, for all j . (cid:3) N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 28
It follows from Claim 5.2 that(5.4) µ p ( L λ ) = lim n →∞ µ p (cid:0) L ⌊ λ n ⌋ / n (cid:1) , where we can regard L ⌊ λ n ⌋ / n either as a subset of P ( N ) (with µ p = µ ( N ) p ) or as asubset of P ([ n ]) (with µ p = µ ( n ) p , the p -biased measure on P ([ n ]) ); the two measurescoincide on families depending only upon the first n coordinates. (Alternatively, itis easy to deduce (5.4) from (5.3).)In order to analyse I p [ L λ ] , we need some further observations. If A ⊂ P ( N ) , wewrite A + i = { S \ { i } : i ∈ S, S ∈ A} ⊂ P ( N \ { i } ) , and we write A − i = { S ∈ A : i / ∈ S } ⊂ P ( N \ { i } ) . If i ∈ N , we define the ‘projected’ σ -algebra Σ i := {{ S \ { i } : S ∈ F} : F ∈ Σ } ⊂ P ( N \ { i } ) , and we equip ( P ( N \ { i } ) , Σ i ) with the natural product measure µ ( N \{ i } ) p inducedby µ ( N ) p , i.e. for all G ∈ Σ i , µ ( N \{ i } ) p ( G ) := µ ( N ) p ( { F ⊂ N : F \ { i } ∈ G} ) . It is easily checked that if
A ∈ Σ , then A + i , A − i ∈ Σ i , and if moreover A is monotoneincreasing, then I pi [ A ] = µ ( N \{ i } ) p ( A + i \ A − i ) = µ ( N \{ i } ) p ( A + i ) − µ ( N \{ i } ) p ( A − i ) . For brevity, we will write µ p = µ ( N \{ i } ) p when the underlying space { , } N \{ i } isclear from the context.We can now prove the following. Claim 5.3.
Let < p < , let ≤ λ ≤ and let i ∈ N . Then I pi [ L ] ≤ r i / (1 − r ) .Proof. Since L = L λ is monotone increasing, we have I pi [ L ] = µ p ( L + i \ L − i ) . If S ∈ L + i \ L − i , then S ∪ { i } ∈ L \ L ( i − , since L ( i − depends only upon thefirst i − coordinates. Since µ ( N ) p ( { S ∪ { i }} ) = pµ ( N \{ i } ) p ( { S } ) for each such S , wehave pµ ( N \{ i } ) p ( L + i \ L − i ) ≤ µ ( N ) p ( L \ L ( i − . By Claim 5.2, we have µ p ( L \ L ( i − ≤ r i / (1 − r ) , and therefore I pi [ L ] = µ ( N \{ i } ) p ( L + i \ L − i ) ≤ µ ( N ) p ( L \ L ( i − p ≤ r i p (1 − r ) ≤ r i (1 − r ) , as required. (cid:3) It follows from Claim 5.3 that I p [ L λ ] ≤ P ∞ i =1 r i / (1 − r ) = r/ (1 − r ) < ∞ , forany p ∈ (0 , and any λ ∈ [0 , . Claim 5.4.
Let < p < and let ≤ λ ≤ . Then for each i ∈ N , we have | I pi [ L ] − I pi [ L ( n )] | ≤ r n (1 − r ) . N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 29
Proof.
Observe that for any monotone increasing A , B ∈ Σ with B ⊂ A , and any i ∈ N , we have | I pi [ A ] − I pi [ B ] | = (cid:12)(cid:12) ( µ p ( A + i ) − µ p ( A − i )) − ( µ p ( B + i ) − µ p ( B − i )) (cid:12)(cid:12) = | ( µ p ( A + i ) − µ p ( B + i )) − ( µ p ( A − i ) − µ p ( B − i )) |≤ max { µ p ( A + i ) − µ p ( B + i ) , µ p ( A − i ) − µ p ( B − i ) } = max { µ p ( A + i \ B + i ) , µ p ( A − i \ B − i ) }≤ µ p ( A \ B )min { p, − p } = µ p ( A \ B )1 − r . Applying this with A = L and B = L ( n ) , and using Claim 5.2, yields | I pi [ L ] − I pi [ L ( n )] | ≤ r n +1 (1 − r ) ∀ i ∈ N , as required. (cid:3) The two claims above yield the following.
Lemma 5.5. | I p [ L ] − I p [ L ( n )] | ≤ nr n (1 − r ) . Proof.
Since L ( n ) depends only upon the first n coordinates, we have I pi [ L ( n )] = 0 for all i > n . Hence, | I p [ L ] − I p [ L ( n )] | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 ( I pi [ L ] − I pi [ L ( n )]) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X i = n +1 ( I pi [ L ] − I pi [ L ( n )]) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 ( I pi [ L ] − I pi [ L ( n )]) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ∞ X i = n +1 I pi [ L ] ≤ n X i =1 | I pi [ L ] − I pi [ L ( n )] | + ∞ X n +1 I pi [ L ] ≤ n r n (1 − r ) + ∞ X i = n +1 r i (1 − r ) = ((1 − r ) n + r ) r n (1 − r ) ≤ nr n (1 − r ) , where the third inequality uses Claim 5.4 to bound the first sum and Claim 5.3 tobound the second. (cid:3) Lemma 5.5 implies that(5.5) I p [ L λ ] = lim n →∞ I p (cid:2) L ⌊ λ n ⌋ / n (cid:3) , where we can regard L ⌊ λ n ⌋ / n either as a subset of P ( N ) or as a subset of P ([ n ]) ;the two relevant notions of influence coincide on families depending only upon thefirst n coordinates.Lemma 5.5 also implies that the statement of the Margulis-Russo lemma holdsfor L λ : N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 30
Lemma 5.6. If < p < and ≤ λ ≤ , then the function p µ p ( L λ ) isdifferentiable at p , with ddp µ p ( L λ ) (cid:12)(cid:12)(cid:12)(cid:12) p = p = I p [ L λ ] . Proof.
We may assume that < λ < . Fix such a λ . Define the function g :(0 , → [0 , g ( p ) = µ p ( L ) , and for each n ∈ N , define a function g n : (0 , → [0 , g n ( p ) = µ p ( L ( n )) . By (5.4), g n ( p ) → g ( p ) as n → ∞ , for any p ∈ (0 , . Bythe Margulis-Russo lemma, g ′ n ( p ) = I p [ L ( n )] for each n ∈ N , since for each n ∈ N ,the family L ( n ) ⊂ P ( N ) can be viewed as a subset of P ([ n ]) , with the respectivedefinitions of total influence coinciding. Moreover, by Lemma 5.5, provided η ≤ p ≤ − η where η > , we have(5.6) | I p [ L ] − g ′ n ( p ) | = | I p [ L ] − I p [ L ( n )] | ≤ n (1 − η ) n η → as n → ∞ , so g ′ n converges uniformly to the function p I p [ L ] on the interval [ η, − η ] , forany η > . It follows from the Differentiable Limit theorem that g is differentiable,and that for any p ∈ (0 , we have ddp µ p ( L ) (cid:12)(cid:12)(cid:12)(cid:12) p = p = g ′ ( p ) = lim n →∞ g ′ n ( p ) = lim n →∞ I p [ L ( n )] = I p [ L ] , using (5.6) again for the last equality. This proves the lemma. (cid:3) We also need the following claims.
Claim 5.7.
Let < p < and let F ∈ Σ . Then µ p ( F ) ≤ ( µ / ( F )) log / ( r ) . Proof.
Let < p < . Since the algebra of sets {F × P ( N \ [ n ]) : n ∈ N , F ⊂ P ([ n ]) } is dense in the probability space ( P ( N ) , Σ , µ p ) and in the probability space ( P ( N ) , Σ , µ / ) ,it suffices to prove the claim when F ⊂ P ([ n ]) for some n ∈ N .Let S ⊂ [ n ] . Then µ p ( { S } ) = p | S | (1 − p ) n −| S | ≤ r n = (2 − n ) log / ( r ) = ( µ / ( { S } )) log / ( r ) . Hence, for any
F ⊂ P ([ n ]) , we have µ p ( F ) = X S ∈F µ p ( { S } ) ≤ X S ∈F ( µ / ( { S } )) log / ( r ) ≤ X S ∈F µ / ( { S } ) ! log / ( r ) = ( µ / ( F )) log / ( r ) , the last inequality using the fact that log / ( r ) ≥ . (cid:3) Claim 5.8.
Let < p < . The function f p : [0 , → [0 , λ µ p ( L λ ) iscontinuous. N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 31
Proof.
Let < p < . Observe that µ / ( L λ ) = λ for all λ ∈ [0 , , and that sincethe families L λ are nested, f p is monotone increasing. Let ≤ λ < λ ′ ≤ . Thefamily L λ ′ \ L λ is clearly Σ -measurable, and we have f p ( λ ′ ) − f p ( λ ) = µ p ( L λ ′ ) − µ p ( L λ ) = µ p ( L λ ′ \ L λ ) ≤ ( µ / ( L λ ′ \ L λ )) log / ( r ) = ( λ ′ − λ ) log / ( r ) → as λ ′ − λ → , using Claim 5.7 for the last inequality. It follows that f p is continuous, as required. (cid:3) We now know that for each p ∈ (0 , , the function f p : λ µ ( N ) p ( L λ ) iscontinuous and monotone increasing, with f p (0) = 0 and f p (1) = 1 . Hence, bythe intermediate value theorem, for any p ∈ (0 , and any x ∈ [0 , , there exists λ ∈ [0 , such that µ ( N ) p ( L λ ) = x . In particular, for each n ∈ N and each F ⊂P ([ n ]) , there exists λ ∈ [0 , such that µ ( N ) p ( L λ ) = µ ( n ) p ( F ) , where µ ( n ) p denotes the p -biased measure on P ([ n ]) , i.e. there always exists a λ ∈ [0 , as in the hypothesisof Theorem 1.9. The Kruskal-Katona theorem, and some applications.
In our proof of The-orem 1.9, we will also use the well-known Kruskal-Katona theorem [26, 29]. Tostate it, we need some more notation. For k, n ∈ N ∪ { } with ≤ k ≤ n , wewrite [ n ] ( k ) := { S ⊂ [ n ] : | S | = k } . For a family F ⊂ P ([ n ]) and ≤ k ≤ n ,we write F ( k ) := F ∩ [ n ] ( k ) . If k < n and A ⊂ [ n ] ( k ) , we write ∂ + ( A ) := { B ∈ [ n ] ( k +1) : A ⊂ B for some A ∈ A} for the upper shadow of A , and if ≤ i ≤ n − k ,we write ∂ +( i ) ( A ) := { B ∈ [ n ] ( k + i ) : A ⊂ B for some A ∈ A} for its i th iterate.We define the lexicographic ordering on [ n ] ( k ) to be the restriction to [ n ] ( k ) of thelexicographic ordering on P ([ n ]) , i.e. if S, T ∈ [ n ] ( k ) , then S > T iff min( S ∆ T ) ∈ S .If ≤ m ≤ (cid:0) nk (cid:1) , we define L ( n,k,m ) to be the size- m initial segment of the lexico-graphic ordering on [ n ] ( k ) , i.e. the m largest elements of [ n ] ( k ) with respect to thelexicographic ordering. Clearly, for any ≤ m ≤ (cid:0) nk (cid:1) , we have L ( n,k,m ) = L ∩ [ n ] ( k ) for some initial segment L of the lexicographic ordering on P ([ n ]) .We can now state the Kruskal-Katona theorem. Theorem 5.9 (Kruskal-Katona theorem) . Let ≤ k < n , and let F ⊂ [ n ] ( k ) .Then | ∂ + ( F ) | ≥ | ∂ + ( L ( n,k, |F| ) | . We need the following straightforward corollary.
Corollary 5.10.
Let n > k > k ≥ j ≥ with n − k ≥ j , suppose that L ⊂ P ([ n ]) is a lexicographically ordered family depending only upon the coordinates in [ j ] ,and let F ⊂ P ([ n ]) be a monotone increasing family with |F ( k ) | ≤ |L ( k ) | . Then |F ( k ) | ≤ |L ( k ) | .Proof. Suppose that |F ( k ) | ≤ |L ( k ) | , and assume for a contradiction that |F ( k ) | ≥|L ( k ) | + 1 . Let ˜ L ⊂ P ([ n ]) be the minimal lexicographically ordered family suchthat |F ( k ) | = | ˜ L ( k ) | ; then ˜ L ( k ) \ L ( k ) = ∅ . Choose S ∈ ˜ L ( k ) \ L ( k ) . Since k ≤ n − j ,there exists S ′ ⊃ S such that | S ′ | = k and ( S ′ \ S ) ∩ [ j ] = ∅ , and therefore S ′ ∈ ∂ +( k − k ) ( ˜ L ( k ) ) \L . Since j ≤ k and L depends only upon the coordinates in [ j ] ,we have L ( k ) = ∂ +( k − k ) ( L ( k ) ) ⊂ ∂ +( k − k ) ( ˜ L ( k ) ) . It follows that | ∂ +( k − k ) ( ˜ L ( k ) ) | > N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 32 |L ( k ) | . By repeated application of the Kruskal-Katona theorem, since |F ( k ) | = | ˜ L ( k ) | and F is monotone increasing, we have |F ( k ) | ≥ | ∂ +( k − k ) ( F ( k ) ) | ≥ | ∂ +( k − k ) ( ˜ L ( k ) ) | > |L ( k ) | , a contradiction. (cid:3) This implies the following, by a standard application of the method of Dinur-Safra[5] / Frankl-Tokushige [13], known as ‘going to infinity and back’. (We present theproof, for completeness.)
Corollary 5.11.
Let < q < p < , let < λ < , and let F ⊂ P ([ n ]) be amonotone increasing family with µ p ( F ) ≤ µ p ( L λ ) . Then µ q ( F ) ≤ µ q ( L λ ) .Proof. Let
F ⊂ P ([ n ]) be a monotone increasing family with µ p ( F ) ≤ µ p ( L λ ) ,and suppose for a contradiction that µ q ( F ) > µ q ( L λ ) . By Claim 5.8, there exists λ ′ > λ such that µ q ( F ) > µ q ( L λ ′ ) . By (5.4), there exists m ≥ n such that µ ( m ) p ( L λ ′ ∩ P ([ m ])) > µ p ( L λ ) , µ ( m ) q ( L λ ′ ∩ P ([ m ])) > µ q ( L λ ) . Define L ′ = L λ ′ ∩ P ([ m ]) ⊂ P ([ m ]) ; then µ p ( L ′ ) > µ p ( L λ ) , µ q ( L ′ ) > µ q ( L λ ) . Now, for any family
G ⊂ P ([ n ]) and any N ∈ N with N ≥ n , we define G N := { A ⊂ [ N ] : A ∩ [ n ] ∈ G} . It is easily checked that for any
G ⊂ P ([ n ]) and any p ∈ (0 , , we have µ p ( G ) = lim N →∞ | ( G N ) ( ⌊ pN ⌋ ) | (cid:0) N ⌊ pN ⌋ (cid:1) . In particular, we have µ q ( F ) = lim N →∞ | ( F N ) ( ⌊ qN ⌋ ) | (cid:0) N ⌊ qN ⌋ (cid:1) and µ q ( L ′ ) = lim N →∞ | ( L ′ N ) ( ⌊ qN ⌋ ) | (cid:0) N ⌊ qN ⌋ (cid:1) . Since µ q ( F ) > µ q ( L λ ′ ) ≥ µ q ( L ′ ) , for all N sufficiently large (depending on q and m ), we have | ( F N ) ( ⌊ qN ⌋ ) | > | ( L ′ N ) ( ⌊ qN ⌋ ) | . Since L ′ N depends only upon the coordinates in [ m ] , and is a lexicographic family,it follows from Corollary 5.10 that if N is sufficiently large depending on p, q and m , then | ( F N ) ( ⌊ pN ⌋ ) | > | ( L ′ N ) ( ⌊ pN ⌋ ) | . Since µ p ( F ) = lim N →∞ | ( F N ) ( ⌊ pN ⌋ ) | (cid:0) N ⌊ pN ⌋ (cid:1) and µ p ( L ′ ) = lim N →∞ | ( L ′ N ) ( ⌊ pN ⌋ ) | (cid:0) N ⌊ pN ⌋ (cid:1) , it follows that µ p ( F ) ≥ µ p ( L ′ ) > µ p ( L λ ) , a contradiction. (cid:3) N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 33
Now we are ready to prove Theorem 1.9.
Proof of Theorem 1.9.
Let F be a family that satisfies the assumptions of the theo-rem. Note that by Lemmas 5.1 and 5.6, for any p ∈ (0 , , we have ddp µ p ( L λ ) | p = p = I p [ L λ ] and ddp µ p ( F ) | p = p = I p [ F ] . By Corollary 5.11, µ q ( F ) ≤ µ q ( L λ ) for any q ≤ p . Therefore, I p [ F ] = lim q → p µ p ( F ) − µ q ( F ) p − q ≥ lim q → p µ p ( L λ ) − µ q ( L λ ) p − q = I p [ L λ ] , as desired. (cid:3) The deduction of Theorem 1.1 from Theorem 1.9.
This is a standard (andshort) ‘monotonization’ argument. We include it for completeness.For i ∈ [ n ] , the i th monotonization operator M i : P ([ n ]) → P ([ n ]) is defined asfollows. (See e.g. [23].) If F ⊂ P ([ n ]) , then for each S ∈ F we define M i ( S ) = ( S ∪ { i } if S ∈ F , i / ∈ S and S ∪ { i } / ∈ F ,S otherwise , and we define M i ( F ) = {M i ( S ) : S ∈ F} . It is well-known, and easy to check,that for any F ⊂ P ([ n ]) , we have |M i ( F ) | = |F| and I / j [ M i ( F )] ≤ I / j [ F ] ∀ j ∈ [ n ]; summing over all j we obtain I / [ M i ( F )] ≤ I / [ F ] . Observe that the M i ’s transform a family to a monotone increasing one, in thesense that for any F ⊂ P ([ n ]) , the family G := M ◦ · · · ◦ M n ( F ) is monotoneincreasing; note also that |G| = |F| and I / [ G ] ≤ I / [ F ] .Now let F ⊂ P ([ n ]) , and let L λ ⊂ P ([ n ]) be a lexicographic family with |L λ | = |F| . Let G = M ◦ · · · ◦ M n ( F ) ; then |G| = |F| , I / [ G ] ≤ I / [ F ] , and G ismonotone increasing. By Theorem 1.9, we have I / [ G ] ≥ I / [ L λ ] , and therefore I / [ F ] ≥ I / [ G ] ≥ I / [ L λ ] , proving Theorem 1.1. Remark 5.12.
We observe that the statement of Theorem 1.1 does not hold forarbitary (i.e., non-monotone) families F , if p = 1 / . Indeed, let F = { S ⊂ [ n ] : 1 / ∈ S } , and let p ∈ (0 , \ { / } ; then µ p ( F ) = 1 − p and I p [ F ] = 1 . Since the function f p : λ µ p ( L λ ) is continuous (by Claim 5.8) with f p (0) = 0 and f p (1) = 1 , thereexists λ ∈ (0 , such that µ p ( L λ ) = 1 − p . Write L = L λ , and as before, for each n ∈ N , write L ( n ) = L ⌊ λ n ⌋ / n . Then we may view L ( n ) as a subset of P ([ n ]) , foreach n ∈ N . We have µ p ( L ( n )) → µ p ( L ) = 1 − p as n → ∞ , by (5.4).First suppose that / < p < . By Theorem 1.6, and since L ( n ) is monotoneincreasing with µ p ( L ( n )) → − p as n → ∞ , we have pI p [ L ( n )] ≥ µ p ( L ( n )) log p ( µ p ( L ( n ))) → (1 − p ) log p (1 − p ) as n → ∞ . It follows from (5.5) that pI p [ L ] ≥ (1 − p ) log p (1 − p ) > p, the last inequality using Claim 2.2 and the fact that p > / . Hence, I p [ L ] > I p [ F ] . N A BIASED EDGE ISOPERIMETRIC INEQUALITY FOR THE DISCRETE CUBE 34
Now suppose that < p < / . Note that L ( n ) ∗ ⊂ P ([ n ]) is monotone increasingwith µ − p ( L ( n ) ∗ ) = 1 − µ p ( L ( n )) and I − p [ L ( n ) ∗ ] = I p [ L ( n )] . By Theorem 1.6,and since L ( n ) ∗ is monotone increasing, we have (1 − p ) I p [ L ( n )] = (1 − p ) I − p [ L ( n ) ∗ ] ≥ µ − p ( L ( n ) ∗ ) log − p ( µ − p ( L ( n ) ∗ )) → p log − p ( p ) as n → ∞ , since µ − p ( L ( n ) ∗ ) = 1 − µ p ( L ( n )) → p as n → ∞ . It follows from (5.5)that (1 − p ) I p [ L ] ≥ p log − p ( p ) > − p, the last inequality using Claim 2.2 and the fact that p < / . Hence, I p [ L ] > I p [ F ] . 6. Open Problems
A natural open problem is to obtain a p -biased edge-isoperimetric inequality forarbitrary (i.e., not necessarily monotone increasing) families, which is sharp for allvalues of the p -biased measure. This is likely to be difficult, as there is no nestedsequence of extremal families. Indeed, it is easily checked that if p < / , the uniquefamilies F ⊂ P ([ n ]) with µ p ( F ) = p and minimal I p [ F ] are the dictatorships,whereas the unique families G ⊂ P ([ n ]) with µ p ( G ) = 1 − p and minimal I p [ G ] arethe antidictatorships; clearly, none of the former are contained in any of the latter.Another natural problem is to obtain a sharp stability version of our ‘full’ biasededge isoperimetric inequality for monotone increasing families (i.e., Theorem 1.9).This would generalise (the monotone case of) Theorem 1.4, our sharp stabilityversion of the ‘full’ edge isoperimetric inequality. It seems likely that the proof in[9] can be extended to the biased case using the methods of the current paper, butthe resulting proof is expected to be rather long and complex.Finally, it is highly likely that the values of the absolute constants in Theorem1.7, and of the constants depending upon η in Theorem 1.8, could be substantiallyimproved. Note for example that Theorem 1.7 applies only to Boolean functionswhose total influence is very close to the minimum possible, namely, for pI p [ f ] ≤ µ p [ f ] (cid:0) log p ( µ p [ f ]) + ǫ (cid:1) , where ǫ ≤ c / ln(1 /p ) and c is very small. It is likely thatthe conclusion holds under the weaker assumption ǫ < / ln(1 /p ) . Such an extensionis not known even for the uniform measure. (See, for example, the conjectures in[6].) References [1] R. Ahlswede and L. H. Khachatrian, The complete intersection theorem for systemsof finite sets,
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David Ellis, School of Mathematical Sciences, Queen Mary, University of London,Mile End Road, London E1 4NS, UK.
E-mail address : [email protected] Nathan Keller, Department of Mathematics, Bar Ilan University, Ramat Gan5290002, Israel.
E-mail address : [email protected] Noam Lifshitz, Department of Mathematics, Bar Ilan University, Ramat Gan5290002, Israel.
E-mail address ::