Approximate convexity and an edge-isoperimetric estimate
aa r X i v : . [ m a t h . F A ] N ov APPROXIMATE CONVEXITYAND AN EDGE-ISOPERIMETRIC ESTIMATE
VSEVOLOD F. LEV
Abstract.
We study extremal properties of the function F ( x ) := min { k k x k − /k : k ≥ } , x ∈ [0 , , where k x k = min { x, − x } . In particular, we show that F is the pointwise largestfunction of the class of all real-valued functions f defined on the interval [0 , f ( λx + (1 − λ ) x ) ≤ λf ( x ) + (1 − λ ) f ( x ) + | x − x | , x , x , λ ∈ [0 , { f (0) , f (1) } ≤ A and S are subsets of a finite abelian group G , such that S is generating and all of its elements have order at most m , thenthe number of edges from A to its complement G \ A in the directed Cayley graphinduced by S on G is ∂ S ( A ) ≥ m | G | F ( | A | / | G | ) . Summary of results
In this section we discuss our principal results; the proofs are presented in Sec-tions 2–4.The central character of this paper is the function F ( x ) := min { k k x k − /k : k ≥ } , x ∈ [0 , , where k runs over positive integers, and k x k denotes the distance from x to the nearestinteger; that is, k x k = min { x, − x } for x ∈ [0 , β = 1 / β k := (1 + 1 /k ) − k ( k +1) for integer k ≥ / β > β > · · · ), we have F ( x ) = kx − /k whenever β k ≤ x ≤ β k − , and F (1 − x ) = F ( x ). The graphs of thefunction F and the functions kx − /k for k ∈ { , , } are presented in Figure 1.Recall that for c, p >
0, a real-valued function f defined on a convex subset of a(real) normed vector space is called ( c, p )-convex if it satisfies f ( λx + (1 − λ ) x ) ≤ λf ( x ) + (1 − λ ) f ( x ) + c k x − x k p Mathematics Subject Classification.
Primary: 39B62; secondary: 26A51, 05C35, 05D99.
Key words and phrases.
Approximate convexity, edge-isoperimetric inequalities.
Figure 1.
The graphs of the functions F and kx − /k for k ∈ { , , } .for all x and x in the domain of f , and all λ ∈ [0 , F the class ofall (1 , , F consists of all real-valuedfunctions f , defined on [0 ,
1] and satisfying f ( λx + (1 − λ ) x ) ≤ λf ( x ) + (1 − λ ) f ( x ) + | x − x | , x , x , λ ∈ [0 , . (1)Also, let F be the subclass of all those functions f ∈ F satisfying the boundarycondition max { f (0) , f (1) } ≤ . (2)Since F is closed under translates by a linear function, any function f ∈ F can beforced into F just by adding to it an appropriate linear summand. Hence, studyingthese two classes is essentially equivalent.It is immediate from the definition that the classes F and F are “symmetric around x = 1 /
2” in the sense that a function f belongs to the class F ( F ) if and only if sodoes the function x f (1 − x ) ( x ∈ [0 , F lies in the class F and indeed, is the (pointwise) largest function of this class. Theorem 1.
We have F ∈ F and f ≤ F for every function f ∈ F . We remark that substituting x = 1 and x = 0 into (1) shows that all functionsfrom the class F are uniformly bounded from above, whence the function sup { f : f ∈F } is well defined. It is not difficult to see that this function itself belongs to F andis pointwise bounded from above by the function F . However, proving that F ∈ F is more delicate. PPROXIMATE CONVEXITY AND AN EDGE-ISOPERIMETRIC ESTIMATE 3
For integer m ≥
2, let F m denote the class of all real-valued functions, defined onthe interval [0 ,
1] and satisfying the boundary condition (2) and the estimate f (cid:18) x + · · · + x m m (cid:19) ≤ f ( x ) + · · · + f ( x m ) m + ( x m − x ) , (3)for all x , . . . , x m ∈ [0 ,
1] with min i x i = x and max i x i = x m . These classes wereintroduced in [L], except that the functions from the class F (up to the normalization(2)) are known as (1 , midconvex and under this name have been studied in anumber of papers; see, for instance, [TT09b]. As shown in [L, Lemma 3], everyconcave function from the class F belongs to all classes F m . Thus, from Theorem 1we get Corollary 1.
We have F ∈ F m for all m ≥ . As an application, we establish a result from the seemingly unrelated realm of edgeisoperimetry.The edge-isoperimetric problem for a graph on the vertex set V is to find, forevery non-negative integer n ≤ | V | , the smallest possible number of edges betweenan n -element set of vertices and its complement in V . We refer the reader to thesurvey of Bezrukov [B96] and the monograph of Harper [H04] for the history, generalperspective, numerous results, variations, and further references on this and relatedproblems.In the present paper we are concerned with the situation where the graph underconsideration is a Cayley graph on a finite abelian group. We use the followingnotation. Given two subsets A, S ⊆ G of a finite abelian group G , we considerthe directed Cayley graph, induced by S on G , and we write ∂ S ( A ) for the edge-boundary of A ; that is, ∂ S ( A ) is number of edges in this graph from an element of A to an element in its complement G \ A : ∂ S ( A ) := |{ ( a, s ) ∈ A × S : a + s / ∈ A }| . Equivalently, ∂ S ( A ) is the number of edges between A and G \ A in the undirected Cayley graph induced on G by the set S ∪ ( − S ).As a consequence of Corollary 1 (thus, ultimately, of Theorem 1), we prove Theorem 2.
Let A and S be subsets of a finite abelian group G , of which S isgenerating. If m is a positive integer such that the order of every element of S doesnot exceed m , then ∂ S ( A ) ≥ m | G | F ( | A | / | G | ) . VSEVOLOD F. LEV
Our proof of Theorem 2 is a variation of the argument used in [L, Theorem 5]where a slightly weaker estimate is established under the stronger assumption that all elements of G have order at most m .In the special particular case where G is a homocyclic group of exponent m , and S ⊆ G is a standard generating subset, Theorem 2 gives a result of Bollob´as andLeader [BL91, Theorem 8].The estimate of Theorem 2, in general, fails to hold for non-abelian groups. Forinstance, if G is the symmetric group of order | G | = 6, and S ⊆ G consists of twoinvolutions, then the Cayley graph induced on G by S is a bi-directional cycle oflength 6. Consequently, for a non-empty proper subset A ⊆ G inducing a path inthis cycle, one has ∂ S ( A ) = 2 < | G | F ( | A | / | G | )(as F ( n/
6) = p / n ∈ { , } and F ( n/
6) = 1 for n ∈ { , , } ).It would be interesting to investigate the sharpness of the estimate of Theorem 2and to determine whether the function F in its right-hand side can be replaced witha larger function.Back to the class F , from Theorem 1 we derive the following result showing that,somewhat surprisingly, any function satisfying (1) must actually satisfy a strongerinequality. Theorem 3.
For any function f ∈ F , and any x , x , λ ∈ [0 , , we have f ( λx + (1 − λ ) x ) ≤ λf ( x ) + (1 − λ ) f ( x ) + F ( λ ) | x − x | . Substituting x = 1, x = 0, and f = F into the inequality of Theorem 3, weconclude that the factor F ( λ ) in the right-hand side is optimal, and the function F cannot be replaced with a larger function.As shown in [L, Theorem 6], for each m ≥
2, the functions F m := sup { f : f ∈ F m } are well-defined and belong themselves to the classes F m , and [L, Theorem 5] gives alower bound for the edge-boundary ∂ S ( A ) in terms of these functions. In this contextit is natural to investigate the infimum inf { F m : m ≥ } . It is somewhat surprisingthat this infimum can be found explicitly, even though the individual functions F m are known for 2 ≤ m ≤ Theorem 4.
We have inf { F m : m ≥ } = F. As mentioned above, [L, Lemma 3] says that every concave function from the class F belongs to all the classes F m . For the proof of Theorem 4 we need the converseassertion, which turns out to be true even with the concavity assumption dropped. PPROXIMATE CONVEXITY AND AN EDGE-ISOPERIMETRIC ESTIMATE 5
Lemma 1. If f ∈ F m for all m ≥ , then f ∈ F . Clearly, every convex function lies in the class F . The last result of our papershows that all “sufficiently flat” concave functions also lie in F ; this complementsin a sense Theorem 1 which implies that every concave function from the class F is“flat”.Let C be the class of all functions, defined and concave on the interval [0 ,
1] andvanishing at the endpoints of this interval.
Theorem 5. If f ∈ C and f ( x ) ≤ x (1 − x ) for all x ∈ [0 , , then f ∈ F . Moreover,the function x (1 − x ) is best possible here in the following sense: if h ∈ C has theproperty that for any function f ∈ C with f ≤ h we have f ∈ F , then h ( x ) ≤ x (1 − x ) for all x ∈ [0 , . We now turn to the proofs of the results discussed above. We prove Theorems 1and 2 in Sections 2 and 3, respectively, and the proofs of Theorems 3, 4, and 5 andLemma 1 are presented in Section 4. Concluding remarks are gathered in Section 5.2.
The proof of Theorem 1
First, we show that for any function f ∈ F , and any real x ∈ [0 ,
1] and integer k ≥
1, one has f ( x ) ≤ k k x k − /k ; this will prove the second assertion of the theorem.By symmetry, x ≤ / x = λ k − and x = 0, we get f ( λ k ) ≤ λf ( λ k − ) + λ k − ;that is, f ( λ k ) λ k ≤ f ( λ k − ) λ k − + 1 λ . Iterating, we obtain f ( λ k ) /λ k ≤ k/λ whence, substituting λ = x /k , f ( x ) ≤ kλ k − = kx − /k , as wanted.We now prove that F satisfies (1), and hence F ∈ F . The proof is based onthe following lemma showing that if a concave and continuous function satisfies (1)whenever x ∈ { , } , then it actually satisfies (1) for all x , x ∈ [0 , Lemma 2.
Suppose that the function f is concave and continuous on the interval [0 , . In order for (1) to hold for all λ, x , x ∈ [0 , , it is sufficient that it holds forall x , λ ∈ [0 , and x ∈ { , } . VSEVOLOD F. LEV
To avoid interrupting the flow of exposition, we proceed with the proof of Theo-rem 1, postponing the proof of Lemma 2 to the end of this section.As Lemma 2 shows, it suffices to establish (1) with f = F and x ∈ { , } . Indeed,the case x = 1 reduces easily to that where x = 0 using the symmetry of F aroundthe point 1 /
2. Thus, x = 0 can be assumed, and in view of F (0) = 0, renaming theremaining variable, we have to prove that F ( λx ) ≤ λF ( x ) + x, x, λ ∈ [0 , . (4)The situation where x ∈ { , } is immediate, and we therefore assume that 0 < x < x ≤ /
2, we find k ≥ F ( x ) = kx − /k ,and notice that F ( λx ) ≤ ( k + 1)( λx ) − / ( k +1) by the definition of the function F ;consequently, it suffices to prove that( k + 1)( λx ) − / ( k +1) ≤ kλx − /k + x. Dividing through by x , substituting t := λ / ( k +1) x − / ( k ( k +1)) , and rearranging theterms gives this inequality the shape kt k +1 − ( k + 1) t k + 1 ≥ , and to complete the proof it remains to notice that the left-hand side factors as( t − ( kt k − + ( k − t k − + · · · + 2 t + 1) . The case 1 / ≤ x < < x ≤ / x ′ := 1 − x ,so that 0 < x ′ ≤ /
2. Assuming that (4) fails to hold, we get F ( λx ) > x and also λF ( x ) < F ( λx ) − x ≤ x ′ , (5)which, in view of F ( x ′ ) = F ( x ), jointly yield F ( x ′ ) x ′ < λ < F ( λx ) λx . Since F ( z ) /z is a decreasing function of z (which is obvious for z ∈ [1 / , F for z ∈ [0 , / x ′ > λx . Thus, λx/x ′ <
1, and recalling that x ′ ≤ /
2, by what we have shownabove, F ( λx ) = F (cid:18) λxx ′ · x ′ (cid:19) ≤ λxx ′ F ( x ′ ) + x ′ . Hence, from the assumption that (4) is false, λF ( x ) + x < λxx ′ F ( x ′ ) + x ′ , PPROXIMATE CONVEXITY AND AN EDGE-ISOPERIMETRIC ESTIMATE 7 which can be written as x − x ′ x ′ λF ( x ) > x − x ′ , contradicting (5).It remains to prove Lemma 2. The proof uses the well-known fact that a strictlyconcave function is unimodal; the specific version we need here is that if f is con-tinuous and strictly concave on the closed interval [ u, v ], then either it is strictlymonotonic on [ u, v ], or there exists w ∈ ( u, v ) such that f is strictly increasing on[ u, w ] and strictly decreasing on [ w, v ]. As a result, the minimum of a function, strictlyconcave on a closed interval, is attained at one of the endpoints of the interval. Wealso need Claim 1.
Suppose that the function f is defined and strictly concave on a closedinterval [ u, v ] . If f is monotonically increasing on [ u, v ] , then its inverse is strictlyconvex. If f is monotonically decreasing on [ u, v ] , then its inverse is strictly concave.Proof of Lemma 2. Suppose first that f is strictly concave on [0 , λ, x ∈ [0 ,
1] and x ∈ { , } ; hance, by symmetry, also for all λ, x ∈ [0 ,
1] and x ∈ { , } .Given x , x ∈ [0 ,
1] with x < x , let k := ( f ( x ) − f ( x )) / ( x − x ), and considerthe auxiliary function g ( x ) := f ( x ) − kx (depending on x and x ). Furthermore, set β := g ( x i ) ( i ∈ { , } ) and B := max { g ( x ) : x ∈ [0 , } . Notice, that g ( x ) = g ( x ),in conjunction with the strict concavity of g , implies B > max { g (0) , g (1) } . For any λ ∈ [0 ,
1] we have then f ( λx + (1 − λ ) x ) − λf ( x ) − (1 − λ ) f ( x )= g ( λx + (1 − λ ) x ) − λg ( x ) − (1 − λ ) g ( x ) ≤ B − β, and therefore it suffices to show that x − x + β ≥ B (6)for any 0 < x < x < β = β ( x , x ) and B = B ( x , x ) defined as above.We now put the reasoning onto its head. Suppose that a real k is fixed so that,if g ( x ) = f ( x ) − kx and B = max { g ( x ) : x ∈ [0 , } , then max { g (0) , g (1) } < B .Let w ∈ (0 ,
1) be defined by g ( w ) = B . By the intermediate value property andmonotonicity of g on each of the intervals [0 , w ] and [ w, β withmax { g (0) , g (1) } ≤ β ≤ B there corresponds then a unique pair ( x , x ) with g ( x ) = g ( x ) = β and 0 ≤ x ≤ w ≤ x ≤
1. As the above argument shows, to complete theproof (under the strict concavity assumption) it suffices to establish (6) with x and x understood as functions of the variable β ranging from max { g (0) , g (1) } to B . Since VSEVOLOD F. LEV x and x are actually inverses of the function g restricted to the appropriate intervals,by Claim 1, x is convex and continuous, and x is concave and continuous, so that x − x + β is concave and continuous and consequently, (6) will follow once we obtainit for β = max { g (0) , g (1) } and also for β = B . The latter case (with equality sign) isimmediate from x ≥ x . For the former case, we notice that if β = max { g (0) , g (1) } ,then x (1 − x ) = 0, whence, having t ∈ [0 ,
1] defined by w = tx + (1 − t ) x , we get x − x + β = x − x + B − (cid:0) g ( tx + (1 − t ) x ) − tg ( x ) − (1 − t ) g ( x ) (cid:1) = x − x + B − (cid:0) f ( tx + (1 − t ) x ) − tf ( x ) − (1 − t ) f ( x ) (cid:1) ≥ B by the assumption of the lemma (and the remark at the very beginning of the proof).Finally, suppose that f is concave but, perhaps, not strictly concave on [0 , ε ∈ (0 ,
1) let f ε ( x ) := ( f ( x ) − εx ) / (1 + ε ) and define∆( x , x , λ ) := f ( λx + (1 − λ ) x ) − λf ( x ) − (1 − λ ) f ( x ) − | x − x | and ∆ ε ( x , x , λ ) := f ε ( λx + (1 − λ ) x ) − λf ε ( x ) − (1 − λ ) f ε ( x ) − | x − x | . A straightforward computations confirms that∆( x , x , λ ) = (1 + ε )∆ ε ( x , x , λ ) + ε | x − x | (cid:0) − λ (1 − λ ) | x − x | (cid:1) . (7)Consequently, if ∆( x , x , λ ) ≤ x , λ ∈ [0 ,
1] and x ∈ { , } , then also∆ ε ( x , x , λ ) ≤ f ε is strictly concave (as it is easyto verify), we conclude that ∆ ε ( x , x , λ ) ≤ x , x , λ ∈ [0 , x , x , λ ) ≤ ε for all x , x , λ ∈ [0 , ε can be chosenarbitrarily small, we have indeed ∆( x , x , λ ) ≤ (cid:3) The proof of Theorem 2
We use induction on | G | . Without loss of generality, we assume that S is a minimal(under inclusion) generating subset. Fix an element s ∈ S and write S := S \ { s } .If S = ∅ , then G is cyclic with | G | being equal to the order of s , whence | G | ≤ m and the assertion follows from F ≤
1. Assuming now that S = ∅ , let H be thesubgroup of G , generated by S ; thus, H is proper and non-trivial. Since the quotientgroup G/H is cyclic, generated by s + H , its order l := | G/H | does not exceed m .For i = 1 , . . . , l set A i := A ∩ ( is + H ) and x i := | A i | / | H | .Fix i ∈ [1 , l ]. By the induction hypothesis (as applied to the subset ( A − is ) ∩ H of the group H with the generating subset S ), the number of edges from an element PPROXIMATE CONVEXITY AND AN EDGE-ISOPERIMETRIC ESTIMATE 9 of A i to an element of ( is + H ) \ A is at least m | H | F ( x i ). Furthermore, the numberof edges from A i to (( i + 1) s + H ) \ A is at leastmax {| A i | − | A i +1 | , } = | H | max { x i − x i +1 , } = 12 | H | (cid:0) | x i − x i +1 | + x i − x i +1 (cid:1) (where x i +1 is to be replaced with x for i = l ). It follows that ∂ S ( A ) ≥ m | H | (cid:0) F ( x ) + · · · + F ( x l ) (cid:1) + 12 | H | (cid:0) | x − x | + · · · + | x l − − x l | + | x l − x | (cid:1) . Choose i, j ∈ [1 , l ] so that x i is the smallest, and x j the largest of the numbers x , . . . , x l . From the triangle inequality, | x − x | + · · · + | x l − − x l | + | x l − x | ≥ x j − x i ) , whence ∂ S ( A ) ≥ m | G | F ( x ) + · · · + F ( x l ) l + | H | ( x j − x i ) ≥ m | G | (cid:18) F ( x ) + · · · + F ( x l ) l + ( x j − x i ) (cid:19) . Recalling that F ∈ F l by Corollary 1, we conclude that ∂ S ( A ) ≥ m | G | F (cid:18) x + · · · + x l l (cid:19) = 1 m | G | F ( | A | / | G | ) , as wanted. 4. The proofs of Lemma 1 and Theorems 3, 4, and 5
Proof of Theorem 3.
Fix x , x ∈ [0 ,
1] with x < x and consider the function ϕ defined by ϕ ( λ ) := f ( λx + (1 − λ ) x ) − λf ( x ) − (1 − λ ) f ( x ) , λ ∈ [0 , . Clearly, we have ϕ (0) = ϕ (1) = 0, and for any u, v, t ∈ [0 ,
1] with u < v , using thefact that f ∈ F , we get ϕ (cid:0) tu +(1 − t ) v (cid:1) = f (cid:0) ( tu + (1 − t ) v ) ( x − x ) + x (cid:1) − ( tu + (1 − t ) v ) ( f ( x ) − f ( x )) − f ( x )= f (cid:0) t ( u ( x − x ) + x ) + (1 − t )( v ( x − x ) + x ) (cid:1) − t ( uf ( x ) + (1 − u ) f ( x )) − (1 − t )( vf ( x ) + (1 − v ) f ( x )) ≤ tf ( u ( x − x ) + x ) + (1 − t ) f ( v ( x − x ) + x ) + ( v − u )( x − x ) − t ( uf ( x ) + (1 − u ) f ( x )) − (1 − t )( vf ( x ) + (1 − v ) f ( x ))= tϕ ( u ) + (1 − t ) ϕ ( v ) + ( v − u )( x − x ) . Hence, ( x − x ) − ϕ ∈ F , and therefore ϕ ≤ ( x − x ) F by Theorem 1; that is, f ( λx + (1 − λ ) x ) − λf ( x ) − (1 − λ ) f ( x ) ≤ F ( λ )( x − x )for all x , x , λ ∈ [0 ,
1] with x < x . (cid:3) Proof of Lemma 1.
Aiming at a contradiction, suppose that f ∈ ∩ m ≥ F m , but f / ∈F . By the latter assumption, there exist λ, x , x ∈ [0 ,
1] with x ≤ x such that f ( λx + (1 − λ ) x ) > λf ( x ) + (1 − λ ) f ( x ) + ( x − x ) . (8)Clearly, we have λ ∈ (0 , x := λx + (1 − λ ) x ∈ (0 , f iscontinuous at x by [L, Lemma 4] (which says that all functions from the classes F m are continuous on (0 , rational λ ∈ (0 ,
1) for which (8)holds true; say, λ = u/m with integer 0 < u < m . Now (8) can be written as f (cid:18) ux + ( m − u ) x m (cid:19) > uf ( x ) + ( m − u ) f ( x ) m + x − x , contradicting the assumption f ∈ F m . (cid:3) Proof of Theorem 4.
Let F := inf { F m : m ≥ } ; our goal, therefore, is to show that F = F . To begin with, we prove that F ∈ F m , m ≥ . (9)To this end, we fix ε > x , . . . , x m ∈ [0 ,
1] with min i x i = x and max i x i = x m ,and show that F (cid:18) x + · · · + x m m (cid:19) ≤ F ( x ) + · · · + F ( x m ) m + ( x m − x ) + ε. (10)As shown in [L], if k and l are integers with k | l , then F l ⊆ F k , and hence F l ≤ F k .It follows that F ( x ) = lim l →∞ F l ! ( x ) , x ∈ [0 , PPROXIMATE CONVEXITY AND AN EDGE-ISOPERIMETRIC ESTIMATE 11 thus, there is an integer l ≥ m such that F l ! ( x i ) ≤ F ( x i ) + ε for each i ∈ [1 , m ]. Since F l ! ∈ F l ! ⊆ F m in view of m | l !, we then get F (cid:18) x + · · · + x m m (cid:19) ≤ F l ! (cid:18) x + · · · + x m m (cid:19) ≤ F l ! ( x ) + · · · + F l ! ( x m ) m + ( x m − x ) ≤ F ( x ) + · · · + F ( x m ) m + ( x m − x ) + ε, establishing (10), and therefore (9).To complete the proof we notice that (9) and Lemma 1 yield F ∈ F , whence, byTheorem 1, F ≤ F. (11)On the other hand, since F ∈ F is concave, by [L, Lemma 3] we have F ∈ F m forevery m ≥
2. It follows that F ≤ F m for every m ≥
2, implying F ≤ F . (12)Comparing (11) and (12), we get the assertion. (cid:3) Proof of Theorem 5.
For the first assertion of the theorem we show that (1) holdstrue, provided that f ∈ C and f ( x ) ≤ x (1 − x ) , x ∈ [0 , x < x and 0 < λ <
1. Furthermore,letting x := λx + (1 − λ ) x , we rewrite the inequality to prove as f ( x ) ≤ x − x x − x f ( x ) + x − x x − x f ( x ) + x − x . (13)By concavity, we have f ( x ) ≤ x x f ( x )(as the point ( x , f ( x )) lies above the segment joining the points (0 ,
0) and ( x , f ( x )),and f ( x ) ≤ − x − x f ( x )(as ( x , f ( x )) is above the segment joining ( x , f ( x )) and (1 , f ( x ) ≤ x (1 − x ) . Comparing with (13), we see that it suffices to prove thatmin (cid:26) x x f ( x ) , − x − x f ( x ) , x (1 − x ) (cid:27) ≤ x − x x − x f ( x ) + x − x x − x f ( x ) + x − x . Assuming for a contradiction that this is wrong, after tedious, but routine algebraicmanipulations we then derive x f ( x ) − x f ( x ) > x ( x − x ) x − x , ( x − f ( x ) + (1 − x ) f ( x ) > (1 − x ) ( x − x ) x − x , and ( x − x ) f ( x ) + ( x − x ) f ( x ) > ( x − x ) − x (1 − x )( x − x ) . Multiplying the first of these inequalities by 1 − x and the second by x , and addingup the resulting estimates and the third inequality, we get4 x (1 − x )( x − x ) > (cid:18) x (1 − x ) x − x + x (1 − x ) x − x + 1 (cid:19) ( x − x ) . It is easily verified that this simplifies to( x − x ) ( x − x )( x − x ) < x − x ( x + x ) + ( x + x ) < , which cannot hold since the left-hand side is a square.To prove the second assertion, suppose that h is a concave function on [0 ,
1] with h (0) = h (1) = 0 and h ( x ) > x (1 − x ) for some x ∈ (0 , f ( x ) := ( h ( x ) x x if 0 ≤ x ≤ x , h ( x )1 − x (1 − x ) if x ≤ x ≤ f is a concave function on [0 ,
1] with f (0) = f (1) = 0, and f ≤ h by concavityof h . We show that, on the other hand, f / ∈ F , and indeed, assuming for definiteness x ≤ /
2, that f ( x ) > f (0) + 12 f (2 x ) + 2 x . To this end we just plug in the definition of f and rewrite this inequality as2 x < h ( x ) − h ( x )1 − x (1 − x ) = h ( x )2(1 − x ) , which is equivalent to the assumption h ( x ) > x (1 − x ). (cid:3) PPROXIMATE CONVEXITY AND AN EDGE-ISOPERIMETRIC ESTIMATE 13 Concluding remarks
It seems natural to extend the class F by considering, for every finite closed interval I and every constant c >
0, the class F ( I, c ) of those real-valued functions f definedon I and satisfying f ( λx + (1 − λ ) x ) ≤ λf ( x ) + (1 − λ ) f ( x ) + c | x − x | for all x , x ∈ I and λ ∈ [0 , f ∈ F ( I, c ) if and only if ( c | I | ) − f ◦ ϕ ∈ F , where ϕ is a linearbijection of [0 ,
1] onto I ; that is, F ( I, c ) is obtained from F by a simple linear scaling.In particular, c | I | F ◦ ϕ − is the largest function of the subclass of all functions from F ( I, c ), non-positive at the endpoints of I . Also, as a corollary of Theorem 3, if f ∈ F ( I, c ), then f ( λx + (1 − λ ) x ) ≤ λf ( x ) + (1 − λ ) f ( x ) + cF ( λ ) | x − x | for all x , x ∈ I and λ ∈ [0 , c, p )-convex functions on a given closedinterval, for every fixed p >
0. Normalization reduces this to studying the class F ( p )0 of all real-valued functions on [0 , f ( λx + (1 − λ ) x ) ≤ λf ( x ) + (1 − λ ) f ( x ) + | x − x | p , for all x , x , λ ∈ [0 , F ( p ) := sup { f : f ∈F ( p ) } is well defined and lies itself in the class F ( p ) , and that for any f ∈ F ( p ) wehave f ( λx + (1 − λ ) x ) ≤ λf ( x ) + (1 − λ ) f ( x ) + F ( p ) ( λ ) | x − x | p , for all x , x , λ ∈ [0 , F ( p ) is the largest function with this property.In view of Theorem 1, one can expect that, perhaps, an explicit expression for thefunctions F ( p ) can be found. Acknowledgement
The author is grateful to Eugen Ionescu for his interest and useful discussions.
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E-mail address : [email protected]@math.haifa.ac.il