Projection inequalities for antichains
Konrad Engel, Themis Mitsis, Christos Pelekis, Christian Reiher
PPROJECTION INEQUALITIES FOR ANTICHAINS
KONRAD ENGEL, THEMIS MITSIS, CHRISTOS PELEKIS, AND CHRISTIAN REIHER
Abstract.
Let n be an integer with n ě
2. A set A Ď R n is called an antichain (resp. weak antichain ) if it does not contain two distinct elements x “ p x , . . . , x n q and y “ p y , . . . , y n q satisfying x i ď y i (resp. x i ă y i ) for all i P t , . . . , n u . We show that theHausdorff dimension of a weak antichain A in the n -dimensional unit cube r , s n is atmost n ´ p n ´ q -dimensional Hausdorff measure of A is at most n , which arethe best possible bounds. This result is derived as a corollary of the following projectioninequality , which may be of independent interest: The p n ´ q -dimensional Hausdorffmeasure of a (weak) antichain A Ď r , s n cannot exceed the sum of the p n ´ q -dimensionalHausdorff measures of the n orthogonal projections of A onto the facets of the unit n -cubecontaining the origin. For the proof of this result we establish a discrete variant of theprojection inequality applicable to weak antichains in Z n and combine it with ideas fromgeometric measure theory. § Introduction
Sperner’s theorem [20], a cornerstone of extremal set theory, determines for each positiveinteger n the maximal size of an antichain in the power set of an n -element set and describesthe extremal configurations. In the statement that follows, r n s denotes the set t , . . . , n u of the first n natural numbers and A Ď ℘ pr n sq is said to be an antichain if x Ę y holds forany distinct x, y P A . Theorem 1.1 (Sperner) . If n ě is an integer and A Ď ℘ pr n sq is an antichain, then | A | ď ` n t n { u ˘ . Equality holds if and only if for some ‘ P (cid:32) t n { u , r n { s ( the set A is thecollection of all ‘ -element subsets of r n s . Sperner’s fundamental result has been generalised in various ways and gave rise to asubstantial body of future developments both within extremal set theory and beyond(see [1, 10]).
Mathematics Subject Classification.
Key words and phrases. antichain; weak antichain; Sperner’s theorem; Hausdorff dimension; Hausdorffmeasure; projection.Research was supported by the Czech Science Foundation, grant number GJ16-07822Y, by GAČRproject 18-01472Y and RVO: 67985840. a r X i v : . [ m a t h . C O ] D ec K. ENGEL, TH. MITSIS, CHR. PELEKIS, AND CHR. REIHER
Observe that via characteristic vectors the power set ℘ pr n sq can be identified with theset t , u n of n -dimensional 0-1-vectors. Moreover, for any two subsets x and y of r n s wehave x Ď y if and only if the characteristic vector of x is coordinate-wise at most thecharacteristic vector of y . Therefore, Sperner’s theorem can be reformulated as a statementabout t , u n equipped with the product partial ordering. It seems natural to ask whathappens when one replaces t , u n by the n -dimensional unit cube r , s n .Let us fix the following notation for discussing such situations. Given two n -tuples x “ p x , . . . , x n q and y “ p y , . . . , y n q in R n , we write x ď y if x i ď y i for all i P r n s .Moreover, if x ď y and x ‰ y we write x ă y , while x * y has the stronger meaning that x i ă y i holds for all i P r n s . A set A Ď R n is called an antichain (resp. weak antichain ) ifit does not contain two elements x and y satisfying x ă y (resp. x * y ). So every antichainis also a weak antichain.In order to get some deeper insight first we replace the unit cube r , s n by its discretiza-tion D nm “ (cid:32) m , m , . . . , m ´ m ( n , where m is a fixed positive integer. De Bruijn et al. [6]proved that the sets A n,‘ “ (cid:32) x P D nm : ř ni “ x i “ ‘ ( , where ‘ P (cid:32) t n p m ´ q{ u , r n p m ´ q{ s ( ,are maximum antichains. Using chains of the form ¨ ¨ ¨ ă p x , x , . . . , x n q ă ` x ` m , x ` m , . . . , x n ` m ˘ ă ¨ ¨ ¨ it is easy to show that the set W n “ t x P D nm : x i “ i u is a maximum weakantichain. Note that | A n,‘ | “ O p m n {? n q as n Ñ 8 and that | W n | “ m n ´ p m ´ q n ,whence | A n,‘ |{| W n | “ O p {? n q as n Ñ 8 .Now we come back to the unit cube r , s n . Obviously, the set A ‹ n “ ! x P r , s n : n ÿ i “ x i “ n { ) is an antichain and the set W ‹ n “ (cid:32) x P r , s n : x i “ i ( is a weak antichain in r , s n . In view of its similarity with the previous extremal config-urations one might expect them to have an interesting maximality property. Questionsaddressing the extremality of (weak) antichains in r , s n become meaningful as soon asone agrees on an (outer) measure on r , s n that allows us to compare any two differentcandidates. The first measure on r , s n that usually comes to mind is the n -dimensionalLebesgue measure. However, the antichain A ‹ n (and also the weak antichain W ‹ n ) is nullwith respect to this measure and the following result due to the first author [9] shows that,actually, all other antichains in r , s n are null in this sense as well. ROJECTION INEQUALITIES FOR ANTICHAINS 3
Theorem 1.2. If c ą and A is a Lebesgue measurable subset of r , s n that does notcontain two elements x ď y with ř ni “ p y i ´ x i q ě c , then the Lebesgue measure of A cannotexceed the Lebesgue measure of the optimal set A p c q “ ! x P r , s n : n ´ c ď n ÿ i “ x i ă n ` c ) . As a matter of fact, the antichain A ‹ n and the weak antichain W ‹ n are not only null withrespect to the Lebesgue measure, but they also have the intuitively stronger property ofbeing p n ´ q -dimensional. One may thus wonder( ) whether every antichain and weak antichain in r , s n is at most p n ´ q -dimensional( ) and if so, whether A ‹ n and W ‹ n are in a natural sense the “largest” p n ´ q -dimensionalantichain resp. weak antichain in r , s n .The perhaps most natural measure theoretic concepts for making these questions preciseare Hausdorff dimension and Hausdorff measure, so let us briefly recall their definitions. If U is a non-empty subset of R n , we denote its diameter by diam p U q . For real numbers s ě δ ą A Ď R n we write H sδ p A q “ α s inf !ÿ i P N diam p U i q s : A Ď ď i P N U i and diam p U i q ď δ for every i P N ) , where the normalisation factor α s “ π s { s Γ p s { ` q denotes the volume of the s -dimensionalsphere of radius . Its presence ensures that if s “ n and t U i : i P N u is a collection ofmutually disjoint balls, then the right side agrees with the total volume of these balls. Forlater use we remark that these quantities H sδ p A q are very robust under the addition ofvarious regularity properties that can be imposed on the sets U i . For instance, one couldinsist that these sets need to be closed (see e.g. [2, p.4]).Evidently for fixed s and A , the value of H sδ p A q increases as δ decreases and thus thelimit H s p A q “ lim δ Ñ H sδ p A q , called the s - dimensional Hausdorff measure of A , exists. It is well-known that H n agreeson R n with the n -dimensional Lebesgue outer measure (see e.g. [11, p.87]). In particular, H n pr , s n q “ Hausdorff dimension of A , denoted by dim H A , is defined bydim H p A q “ inf t s : H s p A q “ u . One checks easily that s ă dim H p A q implies H s p A q “ 8 , while for s ą dim H p A q onehas H s p A q “
0. Therefore, for fixed A the only value of s for which H s p A q can have a“non-trivial” value is s “ dim H p A q . We refer to [2, 11, 12] for legible textbooks on the topic. K. ENGEL, TH. MITSIS, CHR. PELEKIS, AND CHR. REIHER
Let us now return to our problems ( ) and ( ). Our main result does indeed imply thatevery weak antichain (and hence also every antichain) A Ď r , s n satisfies dim H p A q ď n ´ W ‹ n is a union of n facets of the unit n -cube, wherefore H n ´ p W ‹ n q “ n .In order to prove the optimality of W ‹ n we establish a more general result, which we callthe projection inequality . It asserts that the H n ´ -measure of a weak antichain A Ď r , s n is at most the sum of the H n ´ -measures of the orthogonal projections of A to the n facetsof the unit cube containing the origin. Such projections are just deleting a fixed coordinate.When n is clear from the context and i P r n s we write π i : R n ÝÑ R n ´ for the projectiondefined by π i p x , . . . , x n q “ p x , . . . , x i ´ , x i ` , . . . , x n q . Our main result on weak antichains in the unit n -cube reads as follows. Theorem 1.3. If A is a weak antichain in r , s n , then H n ´ p A q ď n ÿ i “ H n ´ ` π i p A q ˘ . In particular, H n ´ p A q ď n and dim H p A q ď n ´ . (1.1)In order to find “better” antichains than A ‹ n one can start with the “best” weak antichain W ‹ n and deform it slightly to obtain an antichain. This can be done in a polyhedral way,but here we present a “smooth” way: Consider the hypersurface A p “ t x P r , s n : } x } p “ u as p Ñ 8 . One can easily verify that the ‘ p -unit ball B p “ t x P R n : } x } p ď u convergeswith respect to the Hausdorff metric to the ‘ -unit ball as p Ñ 8 . Moreover, it is well-known ([19, p.219]) that if a sequence of convex bodies K i converges to a convex body K with respect to the Hausdorff metric, then H n ´ pB K i q Ñ H n ´ pB K q . For these reasons wehave H n ´ p A p q Ñ n as p Ñ 8 . Corollary 1.4.
For every positive integer n we have sup (cid:32) H n ´ p A q : A Ď r , s n is an antichain ( “ n . Note that H p A ‹ q “ ? ă H p A ‹ q “ ? { ă
3, and in general one can easilycheck that H n ´ p A ‹ n q ă n and thus A ‹ n is indeed not optimal. ROJECTION INEQUALITIES FOR ANTICHAINS 5
Obviously, for every antichain A Ď R n , the restriction of π i to A is injective. Weemphasise, however, that the projection inequality does not remain true if the antichain-condition is relaxed to an injectivity-condition. Indeed, Foran [13, p.813] constructedbijective functions from r , s onto r , s whose graphs have arbitrary large H -measure.One of the two central ideas in our approach is to discretise Theorem 1.3. In particular,a basic tool for the proof of Theorem 1.3 is the following discrete variant of the projectioninequality replacing H n ´ by the counting measure. Theorem 1.5. If A is a finite weak antichain in Z n , then | A | ď n ÿ i “ | π i p A q| . Our article is motivated by the idea that several combinatorial statements have continuouscounterparts. This is a rather old idea, which dates back at least to the 70s, and since itsconception many results have been reported in a “measurable” setting (see e.g. [4, 5, 9, 15, 16])or in a “vector space” setting (see e.g. [14, 17]).
Organisation.
In Section 2 we prove Theorem 1.5. We use this result together withsome ideas pertaining to geometric measure theory in Section 3 for proving the specialcase of Theorem 1.3 where A is an antichain. This section might be the technically mostdemanding part of the article and we defer an outline of the argument to Subsection 3.1.In Section 4 we complete the proof of Theorem 1.3 by reducing it to the special case that A is an antichain. The concluding remarks in Section 5 describe several problems for futureresearch. § The discrete projection inequality
The key observation on which the proof of Theorem 1.5 relies is that every weak antichainin Z n can be partitioned into n parts such that for each i P r n s the projection π i p¨q isinjective on the i -th part. Proof of Theorem 1.5.
Let A Ď Z n be a weak antichain. Define recursively disjoint sub-sets A , . . . , A n of A as follows. If for some i P r n s the sets A , . . . , A i ´ have justbeen constructed, set B i “ A (cid:114) p A Y . . . Y A i ´ q and let A i be the set of pointsin B i whose i -th coordinates are minimal. In other words, A i is defined so as to con-tain those points p x , . . . , x n q P B i for which there exists no integer y i ă x i such that p x , . . . , x i ´ , y i , x i ` , . . . , x n q P B i .We contend that the set B n ` “ A (cid:114) p A Y . . . Y A n q is empty. Indeed, assume indirectlythat some point x “ p x , . . . , x n q belongs to B n ` . Using the definitions of A n , . . . , A in K. ENGEL, TH. MITSIS, CHR. PELEKIS, AND CHR. REIHER this order we find integers y n ă x n , . . . , y ă x such that p x , . . . , x i ´ , y i , . . . , y n q P A i forevery i P r n s . In particular, the point y “ p y , . . . , y n q is in A and satisfies y * x , contraryto A being a weak antichain. This proves that A “ A Y¨ A Y¨ . . . Y¨ A n is a partition.Now obviously for every i P r n s the projection π i is injective on A i , whence | A i | “ | π i p A i q| ď | π i p A q| . We conclude that | A | “ n ÿ i “ | A i | ď n ÿ i “ | π i p A q| , as desired. (cid:3) § Antichains
Overview.
This section deals with a special case of our main theorem, where ratherthan weak antichains we only consider antichains. So explicitly we aim at the followingresult.
Proposition 3.1. If A is an antichain in r , s n , then H n ´ p A q ď n ÿ i “ H n ´ ` π i p A q ˘ . As the proof of this estimate is somewhat involved, we would like to devote the presentsubsection to a discussion of our basic strategy. An obvious approach is the followingdiscretisation procedure: Take a large natural number m , cut the unit cube into m n smallercubes of side length m and keep track which of these cubes intersect A . This situationcan be encoded as a weak antichain in r m s n , to which the discrete projection inequality(Theorem 1.5) applies. On first sight one might hope that in the limit m Ñ 8 this argumentwould yield Proposition 3.1. But when working out the details, one discovers that onelooses a constant factor which depends on the dimension n , but not on the antichain A itself. Lemma 3.2.
For every positive integer n there exists a constant D ą such that everyweak antichain A Ď r , s n satisfies H n ´ p A q ď D ¨ n ÿ i “ H n ´ ` π i p A q ˘ . To aid the reader’s orientation we remark that in Subsection 3.2 we will show thisestimate for D “ n p n ´ q{ ¨ α n ´ “ c π n ¨ ´ πe ¯ n { p ` o p qq , (3.1) ROJECTION INEQUALITIES FOR ANTICHAINS 7 but the precise value of D will be rather immaterial to what follows.A completely different and more analytical approach to Proposition 3.1 starts from thefollowing observation. Suppose that f : r , s n ´ ÝÑ r , s is decreasing in each coordinateand sufficiently smooth, and that we want to study the antichain A “ (cid:32) p x , f p x qq : x P r , s n ´ ( . Denoting the partial derivatives of f by D f, . . . , D n ´ f one checks easily that H n ´ p A q “ ż r , s n ´ a ` | D f p x q| ` . . . ` | D n ´ f p x q| d x and H n ´ ` π i p A q ˘ “ ż r , s n ´ ˇˇ D i f p x q ˇˇ d x for all i P r n ´ s , for which reason the projection inequality for A follows from a ` | D f p x q| ` . . . ` | D n ´ f p x q| ď ` | D f p x q| ` . . . ` | D n ´ f p x q| and from π n p A q “ r , s n ´ .In general we need to look at antichains of the form A “ (cid:32) p x , f p x qq : x P B ( , (3.2)where B “ π n p A q Ď r , s n ´ is arbitrary and f : B ÝÑ r , s is only known to be decreasingin each coordinate, but not necessarily smooth. The question to what extent argumentsthat work well for smooth functions can be extended to more general scenarios lies at thevery heart of a mathematical area known as geometric measure theory . In Subsection 3.4we shall use some methods from this subject in order to generalise the previous idea asfollows. Lemma 3.3.
Given an antichain A Ď r , s n and δ ą there exists a Borel set B Ď r , s n ´ such that ( i ) H n ´ p B q ą ´ δ ( ii ) and the set A “ A X π ´ n p B q satisfies H n ´ p A q ď ř ni “ H n ´ ` π i p A q ˘ ` δ. Now roughly speaking one may hope to prove Proposition 3.1 (up to an arbitrarily smalladditive error) by using Lemma 3.3 in each coordinate direction, each time cutting out asubstantial piece of A whose H n ´ -measure can be estimated quite efficiently by part ( ii ).There will remain a “small” left-over part of A , which can then be handled by means ofLemma 3.2. K. ENGEL, TH. MITSIS, CHR. PELEKIS, AND CHR. REIHER
There is one final technical hurdle one needs to overcome when pursuing such a plan.The problem is that, in general, Hausdorff measure is only known to be subadditive. Sowhen one attempts to prove the projection inequality for an antichain A by splitting itinto two pieces, handling both pieces separately, and adding up the results, one can getinto trouble with the right sides. We shall use the following lemma for getting around thisdifficulty. Lemma 3.4.
Let A Ď r , s n be an antichain and let r , s n ´ “ B Y¨ B be a partitionof the p n ´ q -dimensional unit cube into Borel sets. Setting A “ A X π ´ n p B q and A “ A X π ´ n p B q we have H n ´ ` π i p A q ˘ “ H n ´ ` π i p A q ˘ ` H n ´ ` π i p A q ˘ for every i P r n s . We conclude the present subsection by showing that our three lemmas do indeed implyProposition 3.1. The proofs of the lemmas themselves are deferred to the three subsectionsthat follow. But it will be convenient to prove Lemma 3.4 in Subsection 3.3 before we turnour attention to Lemma 3.3 in Subsection 3.4.
Proof of Proposition 3.1 assuming Lemma 3.2–3.4.
Fix a dimension n ě δ ą D be the number provided by Lemma 3.2. We call a subset I Ď r n s good ifevery antichain A Ď r , s n with H n ´ p π i p A qq ď δ for all i P I satisfies H n ´ p A q ď n ÿ i “ H n ´ ` π i p A q ˘ ` p nD ` n ´ | I |q δ . As a consequence of Lemma 3.2 we know that the set r n s is good. Hence there exists aminimal good set I Ď r n s . Assume for the sake of contradiction that I ‰ ∅ . By permutingthe coordinates if necessary we may suppose that n P I . Consider any antichain A Ď r , s n with H n ´ ` π i p A q ˘ ď δ for all i P I (cid:114) t n u . (3.3)By Lemma 3.3 there exists a Borel set B with H n ´ p B q ą ´ δ such that the set A “ A X π ´ n p B q satisfies H n ´ p A q ď n ÿ i “ H n ´ ` π i p A q ˘ ` δ . (3.4)Set B “ r , s n ´ (cid:114) B and A “ A X π ´ n p B q . Notice that H n ´ p π i p A qq ď δ for all i P I .This is because for i P I (cid:114) t n u we have H n ´ ` π i p A q ˘ ď H n ´ ` π i p A q ˘ ď δ , ROJECTION INEQUALITIES FOR ANTICHAINS 9 while for i “ n the first property of B yields H n ´ ` π n p A q ˘ ď H n ´ p B q “ ´ H n ´ p B q ď δ . Since I is good, it follows that H n ´ p A q ď n ÿ i “ H n ´ ` π i p A q ˘ ` p nD ` n ´ | I |q δ . Adding (3.4) and taking Lemma 3.4 into account, we deduce H n ´ p A q ď H n ´ p A q ` H n ´ p A qď n ÿ i “ ` H n ´ ` π i p A q ˘ ` H n ´ ` π i p A q ˘˘ ` p nD ` n ´ | I |q δ ` δ “ n ÿ i “ H n ´ ` π i p A q ˘ ` p nD ` n ´ | I (cid:114) t n u|q δ . As this argument applies to any antichain A with (3.3), we have thereby shown that theset I (cid:114) t n u is good as well, contrary to the minimality of I .This contradiction proves that I “ ∅ is good, or in other words that for every antichain A Ď r , s n the estimate H n ´ p A q ď n ÿ i “ H n ´ ` π i p A q ˘ ` n p D ` q δ holds. In the limit δ Ñ (cid:3) Discretisation.
This subsection deals with the proof of Lemma 3.2. Let us recall thefollowing well-known continuity property of the Hausdorff measure (see [18, Theorem 1.4],the remarks that follow, and [18, Corollary 4.5]).
Fact 3.5. If m ě k ě are integers, E Ď E Ď ¨ ¨ ¨ Ď R m are arbitrary sets, and E “ Ť ‘ ě E ‘ , then H k p E q “ lim ‘ Ñ8 H k p E ‘ q . l The discretisation procedure by means of which we shall establish Lemma 3.2 worksas follows. Fix a large positive integer m the size of which determines how fine ourapproximation is going to be. Let r , s “ I Y¨ . . . Y¨ I m be a partition of the unit intervalinto m consecutive intervals of length m . It is somewhat immaterial how we proceed withthe boundary points m , . . . , m ´ m , but for definiteness we set I j “ $&%“ j ´ m , jm ˘ if j P r m ´ s , “ m ´ m , ‰ if j “ m. This gives rise to the partition r , s n “ ď ¨ d Pr m s n C p d q of the n -dimensional unit cube into m n subcubes defined by C p d q “ I d ˆ ¨ ¨ ¨ ˆ I d n for every d “ p d , . . . , d n q P r m s n .For any subset W Ď r , s n (not necessarily an antichain) we set G m p W q “ (cid:32) d P r m s n : C p d q X W ‰ ∅ ( and observe that the disjoint union H m p W q “ ď ¨ d P G m p W q C p d q (3.5)is a superset of W . Conversely, every point in H m p W q has at most the distance ? n { m ,the common diameter of our small cubes, from an appropriate point in W .Let us introduce some useful notation for such situations. Given S Ď R n and a point x P R n , we set dist p x , S q “ inf t} x ´ s } : s P S u , where } ¨ } denotes the Euclidean norm.For a given positive real number δ and S Ď R n the δ -neighbourhood of S is defined by S p δ q “ t x P R n : dist p x , S q ď δ u . (3.6)Summarising the above discussion, we have W Ď H m p W q Ď W p? n { m q for every W Ď r , s n and every m P N . In the special case where W is closed we have W “ Ş m ě W p? n { m q and, consequently, Fact 3.5 yieldslim m Ñ8 | G m p W q| m n “ H n p W q . (3.7) Lemma 3.6. If A , . . . , A n Ď r , s n ´ are closed sets and A Ď r , s n is a weak antichainwith π i p A q Ď A i for all i P r n s , then H n ´ p A q ď D ¨ n ÿ i “ H n ´ p A i q , where D denotes the constant introduced in (3.1) .Proof. Fix δ ą
0, consider an arbitrary positive integer m ě ? n { δ , and set B “ G m p A q .The covering A Ď Ť d P B C p d q of A uses | B | cubes of diameter ? n { m ď δ and thus we have H n ´ δ p A q ď α n ´ ¨ | B | ¨ ˆ ? nm ˙ n ´ “ D | B | m n ´ . ROJECTION INEQUALITIES FOR ANTICHAINS 11
Since A is a weak antichain in r , s n , the set B is a weak antichain in Z n and Theorem 1.5discloses | B | ď n ÿ i “ | π i p B q| . Obviously, π i p B q Ď G m p A i q for every i P r n s (where the operator G m is applied to the p n ´ q -dimensional unit cube). Hence H n ´ δ p A q ď D ¨ n ÿ i “ | π i p B q| m n ´ ď D ¨ n ÿ i “ | G m p A i q| m n ´ and by (3.7) we obtain in the limit m Ñ 8 that H n ´ δ p A q ď D ¨ n ÿ i “ H n ´ p A i q . Finally δ Ñ (cid:3) Now a standard application of Fact 3.5 leads to Lemma 3.2.
Proof of Lemma 3.2.
Let ε ą i P r n s a sequence p C i,k q k ě ofclosed subsets of r , s n ´ such that ÿ k H n ´ p C i,k q ď H n ´ ` π i p A q ˘ ` ε (3.8)and π i p A q Ď Ť k C i,k .Now consider an arbitrary ‘ ě
1. Setting A p ‘ q i “ ď k Pr ‘ s C i,k for every i P r n s and A p ‘ q “ č i Pr n s π ´ i ` A p ‘ q i ˘ we deduce from Lemma 3.6 that H n ´ p A X A p ‘ q q ď D ¨ n ÿ i “ H n ´ ` A p ‘ q i ˘ . Using A Ď Ť ‘ ě A p ‘ q , Fact 3.5, and (3.8) we obtain in the limit ‘ Ñ 8 that H n ´ p A q ď D ¨ n ÿ i “ H n ´ ` π i p A q ˘ ` nDε . As ε ą (cid:3) Corollary 3.7. If A Ď r , s n is a weak antichain, then H n ´ p A q ď Dn is finite and,consequently, H n p A q “ . l The decomposition Lemma.
Let an n -dimensional antichain A Ď r , s n be given,where n ě
2. Since no two points in A can agree in their first n ´ f A : π n p A q ÝÑ r , s such that A “ (cid:32) p x , f A p x qq : x P π n p A q ( . The fact that A is indeed an antichain is equivalent to f p x q ą f p y q whenever x ă y arein π n p A q . It is often convenient to extend this function f A in a monotonicity preserving wayto the whole p n ´ q -dimensional unit cube. To this end one defines ˆ f A : r , s n ´ ÝÑ r , s by ˆ f A p x q “ inf (cid:32) f A p a q : a P π n p A q and a ď x ( (3.9)for all x P r , s n ´ , where, in this context, inf p ∅ q “
1. By the aforementioned fact on f A we have ˆ f A p x q “ f A p x q for all x P π n p A q . Moreover, if x ď y are in r , s n ´ , then f p x q ě f p y q . We shall refer to ˆ f A as the function associated with the antichain A .More generally, we call a function f : r , s n ´ ÝÑ r , s order-reversing , if we have f p x q ě f p y q whenever x ď y . So for instance the function ˆ f A associated with anantichain A has just been observed to be order-reversing. We will need the followingproperties of such functions proved in [7]. Lemma 3.8. If f : r , s n ´ ÝÑ r , s is order-reversing, then it is measurable in the sensethat preimages of Borel sets are Lebesgue measurable. Moreover, f is almost everywheredifferentiable. Let us record an easy consequence.
Fact 3.9.
Given an antichain A Ď r , s with associated function ˆ f A : r , s n ´ ÝÑ r , s and c P r , s , the set L “ (cid:32) p x , . . . , x n ´ q P r , s n ´ : x n ´ ă ˆ f A p x , . . . , x n ´ , c q ( is Lebesgue measurable.Proof. By Lemma 3.8 it suffices to check that the characteristic function L of L is order-reversing. So let p x , x n ´ q ď p y , y n ´ q be given, where x , y P r , s n ´ . We need to provethat L p x , x n ´ q ě L p y , y n ´ q . If p y , y n ´ q R L this is clear, so we may suppose that p y , y n ´ q P L . Since ˆ f A is order-reversing, it follows that x n ´ ď y n ´ ă ˆ f A p y , c q ď ˆ f A p x , c q , which in turn implies that p x , x n ´ q P L . (cid:3) ROJECTION INEQUALITIES FOR ANTICHAINS 13
Now we are ready for the proof of Lemma 3.4, which will occupy the remainder of thissubsection.
Proof of Lemma 3.4.
For i “ n this follows from the Lebesgue measurability of the set B and from the fact that in R n ´ the p n ´ q -dimensional Hausdorff outer measure coincideswith the p n ´ q -dimensional Lebesgue outer measure. So without loss of generality wemay henceforth assume that i “ n ´ ν : ℘ pr , s n ´ q ÝÑ r , s by setting ν p E q “ H n ´ ` π n ´ p π ´ n p E q X A q ˘ for every E Ď r , s n ´ . In other words, if ˆ f A : r , s n ´ ÝÑ r , s is the function associatedwith the antichain A , then ν p E q “ H n ´ p F E q , where F E “ (cid:32) p x , . . . , x n ´ , ˆ f A p x , . . . , x n ´ qq : p x , . . . , x n ´ q P E X π n p A q ( . Notice that ν is an outer measure. We will show later that B is ν -measurable. This willimply that ν ` π n p A q ˘ “ ν ` π n p A q X B ˘ ` ν ` π n p A q X B ˘ , which is equivalent to H n ´ ` π n ´ p A q ˘ “ H n ´ ` π n ´ p A q ˘ ` H n ´ ` π n ´ p A q ˘ , and the result will follow.Thus it remains to show that all Borel sets B are ν -measurable. It is well known that thesigma algebra of Borel subsets of r , s n ´ is generated by the closed half-spaces bounded byhyperplanes which are orthogonal to the coordinate axes. Therefore it suffices to establishthat for all c P r , s and i P r n ´ s the set B i p c q “ (cid:32) p x , . . . , x n ´ q : x i ď c ( is ν -measurable. This means that for each test set E Ď r , s n ´ we need to prove (see[3, Proposition 1.5.11]) that ν p E q “ ν ` E X B i p c q ˘ ` ν ` E (cid:114) B i p c q ˘ . (3.10)In case i P r n ´ s this rewrites as H n ´ p F E q “ H n ´ ` F E X B i p c q ˘ ` H n ´ ` F E (cid:114) B i p c q ˘ and follows from the measurability of B i p c q . Thus we may suppose i “ n ´ F ´ E “ (cid:32) p x , . . . , x n ´ , ˆ f A p x , . . . , x n ´ qq : p x , . . . , x n ´ q P E X π n p A q and x n ´ ď c ( , and F ` E “ (cid:32) p x , . . . , x n ´ , ˆ f A p x , . . . , x n ´ qq : p x , . . . , x n ´ q P E X π n p A q and x n ´ ą c ( we can reformulate (3.10) as H n ´ p F E q “ H n ´ p F ´ E q ` H n ´ p F ` E q . (3.11)Now by Fact 3.9 the set L “ (cid:32) p x , . . . , x n ´ q P r , s n ´ : x n ´ ă ˆ f A p x , . . . , x n ´ , c q ( is Lebesgue measurable, whence H n ´ p F E q “ H n ´ p F E (cid:114) L q ` H n ´ p F E X L q . (3.12)Moreover, the set N “ (cid:32) p x , . . . , x n ´ , ˆ f A p x , . . . , x n ´ , c qq : x , . . . , x n ´ P r , s ( is a weak antichain in r , s n ´ , so by Corollary 3.7 we have H n ´ p N q “ p F E (cid:114) L q (cid:114) N Ď F ´ E Ď F E (cid:114) L and F E X L Ď F ` E Ď p F E X L q Y N , which follow from the fact that ˆ f A is order-reversing, this shows H n ´ p F ´ E q “ H n ´ p F E (cid:114) L q and H n ´ p F ` E q “ H n ´ p F E X L q . Therefore (3.12) implies (3.11). (cid:3)
Proof of Lemma 3.3.
There are two issues that need to be addressed when transfer-ring the proof of the projection inequality for smooth antichains sketched in Subsection 3.1to the general case. Starting with the representation (3.2) of a given antichain A with anorder-reversing function f , we need to deal with the fact that B may fail to be measurableand, moreover, with the possible non-differentiability of f . It turns out that these twopoints can be handled separately from each other and we start by giving an argument thatapplies to the case where f is linear and B may be arbitrary. Lemma 3.10. If L : R n ´ ÝÑ R is a linear function, B Ď r , s n ´ is arbitrary and S “ (cid:32) p x , L p x qq : x P B ( , then H n ´ p S q ď H n ´ p B q ` n ´ ÿ i “ H n ´ ` π i p S q ˘ . ROJECTION INEQUALITIES FOR ANTICHAINS 15
Proof.
Let L be given by p x , . . . , x n ´ q ÞÝÑ n ´ ÿ i “ c i x i . For i P r n ´ s the map: B ÝÑ π i p S q given by p x , . . . , x n ´ q ÞÝÑ p x , . . . , x i ´ , x i ` , . . . , x n ´ , c x ` . . . ` c n ´ x n ´ q and the map: B ÝÑ S given by p x , . . . , x n ´ q ÞÝÑ p x , . . . , x n ´ , c x ` . . . ` c n ´ x n ´ q are linear and surjective. Since the p n ´ q -dimensional Hausdorff measure of B agreeswith the p n ´ q -dimensional Lebesgue outer measure of B , we have (see e.g. [11, p.114]) H n ´ ` π i p S q ˘ “ | c i | ¨ H n ´ p B q and H n ´ p S q “ ` ` ř n ´ i “ c i ˘ { H n ´ p B q . Thus it remainsto remark ` ` n ´ ÿ i “ c i ˘ { ď ` n ´ ÿ i “ | c i | , which is clear. (cid:3) Recall that a function f : F Ď R n ÝÑ R m is Lipschitz with constant K (or K - Lipschitz for short) if } f p x q ´ f p y q} ď K ¨ } x ´ y } for all x , y P F .
We use several times the following well-known result concerning the s -dimensional Hausdorffmeasure (see [12, p.24]). Lemma 3.11.
Let m and n be positive integers and let F Ď R n . If f : F ÝÑ R m is a K -Lipschitz function, then H s p f p F qq ď K s ¨ H s p F q . For the rest of this subsection we fix an antichain A in r , s n and a positive real number δ for which we would like to establish Lemma 3.3. Let ˆ f A be the function associated with A (see (3.9)). If for some x P p , q n ´ and i P r n ´ s the i -th partial derivative of ˆ f A at x exists, we denote it by D i ˆ f A p x q . Furthermore, if a point x has the property that all partialderivatives D ˆ f A p x q , . . . , D n ´ ˆ f A p x q exist, we define L x : R n ´ ÝÑ R to be the linear formgiven by L x p v , . . . , v n ´ q “ n ´ ÿ i “ D i ˆ f A p x q v i . The Borel set B we need to exhibit will be a subset of a closed set C Ď p , q n ´ onwhich ˆ f A has some useful differentiability properties collected in the lemma that follows. Lemma 3.12.
There exists a closed set C Ď p , q n ´ such that ( i ) H n ´ p C q ą ´ δ { ; ( ii ) all partial derivatives of ˆ f A exist and are continuous on C ; ( iii ) for every x P C , the function ˆ f A is differentiable at x with the derivative L x ; ( iv ) the differentiability of ˆ f A is uniform on C , i.e., for every η ą there exists an ε ą such that for all a , x P C with } a ´ x } ă ε we have ˇˇ ˆ f A p a q ´ ˆ f A p x q ´ L x p a ´ x q ˇˇ ď η } a ´ x } . Proof.
Since the function ˆ f A is order-reversing, Lemma 3.8 implies that it is almosteverywhere differentiable. Hence there exists a measurable set C Ď p , q n ´ , whosemeasure equals 1, such that for every x P C all partial derivatives of ˆ f A exist and,moreover, ˆ f A is differentiable at x with the derivative L x . So by choosing C Ď C later,we can ensure ( iii ) as well as the first part of ( ii ).Next, by Lusin’s theorem (see e.g. [3, Theorem 2.2.10]), there exists a closed set C Ď C with H n ´ p C q ą ´ δ { D i ˆ f A exist and are continuouson C . So every C Ď C satisfies ( ii ) as well.Now define for every m P N the measurable function g m : C ÝÑ R by g m p x q “ sup " | ˆ f A p a q ´ ˆ f A p x q ´ L x p a ´ x q|} a ´ x } : a P ` Q X p , q ˘ n ´ , ă } a ´ x } ă { m * . Since lim m Ñ8 g m p x q “ x P C , Egoroff’s theorem (see e.g. [3, Theo-rem 2.2.1]) implies that there exists a closed set C Ď C with H n ´ p C q ą ´ δ { g m Ñ C . Such a set has the properties ( i ) and ( iv ) as well. (cid:3) Throughout the remainder of this subsection, C denotes a set provided by the previouslemma. Set K “ ˆ ` δn ˙ {p n ´ q (3.13)and for x P p , q n ´ and ε ą Q ε p x q “ (cid:32) y P R n ´ : } x ´ y } ă ε ( be the ε -cube around x . Next we intend to show for every x P C , that if ε ą A X π ´ n p C X Q ε p x qq instead of A . Once this is known, a Vitali covering argument will allow us to combinemany such cubes, so that the desired set B can be taken to be a disjoint union of severalsets of the form C X Q ε p x q . The definition that follows collects some properties of suchcubes that will be useful for implementing this strategy. ROJECTION INEQUALITIES FOR ANTICHAINS 17
Definition 3.13.
Given x P C and ε ą ε -cube Q “ Q ε p x q is said to be nice if it hasthe following properties:( a ) Q Ď r , s n ´ .( b ) If a , b P Q X C , then ˇˇ ˆ f A p a q ´ ˆ f A p b q ˇˇ ď p K ´ q ¨ } a ´ b } ` K ¨ ˇˇ L x p a ´ b q ˇˇ . ( c ) If a , b P Q X C , i P r n ´ s , and D i ˆ f A p x q ‰
0, then ˇˇ L x p a ´ b q ˇˇ ď p K ´ q ÿ j Pr n ´ s (cid:114) t i u | a j ´ b j | ` K ¨ ˇˇ ˆ f A p a q ´ ˆ f A p b q ˇˇ . The following result shows that nice cubes determine parts of A , for which the projectioninequality holds up to a multiplicative factor that is close to 1. Lemma 3.14. If Q “ Q ε p x q is a nice cube and A Q “ A X π ´ n p Q X C q , then H n ´ p A Q q ď K p n ´ q ¨ n ÿ i “ H n ´ ` π i p A Q q ˘ . Proof.
Observe that Definition 3.13( b ) asserts that the map ` a , L x p a q ˘ ÞÝÑ ` a , ˆ f A p a q ˘ from the set S “ (cid:32) p a , L x p a qq : a P π n p A Q q ( onto the set A Q is Lipschitz with constant K .Therefore Lemma 3.11 and Lemma 3.10 (applied with π n p A q and L x here in place of B and L there) yield H n ´ p A Q q ď K n ´ ¨ H n ´ p S qď K n ´ ´ H n ´ ` π n p A Q q ˘ ` n ´ ÿ i “ H n ´ ` π i p S q ˘¯ . So to conclude the proof it suffices to show H n ´ ` π i p S q ˘ ď K n ´ H n ´ ` π i p A Q q ˘ for all i P r n ´ s . (3.14)If D i ˆ f A p x q “
0, the set π i p S q is contained in an p n ´ q -dimensional vector spaceand (3.14) is clear. On the other hand, if D i ˆ f A p x q ‰
0, then Definition 3.13( c ) impliesthat the map ` a i , ˆ f A p a q ˘ ÞÝÑ ` a i , L x p a q ˘ from π i p A Q q to π i p S q is Lipschitz with constant K , which entails (3.14) in view ofLemma 3.11. (cid:3) Next we show that nice cubes are ubiquitous.
Lemma 3.15.
Given x P C , the cube Q ε p x q is nice for every sufficiently small ε ą . Proof.
We verify for each of the three clauses in Defintion 3.13 separately that it holds forevery sufficiently small ε ą
0. Since C Ď p , q n ´ , this is immediate for ( a ). For ( b ), weput η “ K ´ K and let ε ą a , b P Q ε p x q X C Lemma 3.12 yields ˇˇ ˆ f A p a q ´ ˆ f A p b q ˇˇ ( iv ) ď | L a p a ´ b q| ` η } a ´ b }ď | L x p a ´ b q| ` p} L x ´ L a } ` η q} a ´ b } ( ii ) ď | L x p a ´ b q| ` η } a ´ b } . Now the Cauchy-Schwarz inequality implies ˇˇ ˆ f A p a q ´ ˆ f A p b q ˇˇ ď ˆ K ¨ K ¨ | L x p a ´ b q| ` η ? K ´ ¨ ? K ´ ¨ } a ´ b } ˙ ď ˆ K ` η K ´ ˙ ¨ ` K ¨ | L x p a ´ b q| ` p K ´ q ¨ } a ´ b } ˘ , and, as the first factor is equal to 1 by the definition of η , this proves part ( b ) ofDefinition 3.13.It remains to check ( c ). Without loss of generality, we may assume that i “ n ´ D n ´ f p x q ‰
0. Set λ “ ˆ n ´ ÿ i “ D i ˆ f A p x q ˙ { ,λ “ ˇˇ D n ´ ˆ f A p x q ˇˇ ,ξ “ min " λ , λ ? K ´ p λ ` λ q * ,η “ p K ´ q ξ K , let ε ą a , b P Q ε p x q X C . Recall that wehave to show ˇˇ L x p a ´ b q ˇˇ ď p K ´ q n ´ ÿ i “ p a i ´ b i q ` K ¨ ˇˇ ˆ f A p a q ´ ˆ f A p b q ˇˇ . To this end, it suffices to establish the following two implications:( ) If | L x p a ´ b q| ě ξ ¨ } a ´ b } , then | L x p a ´ b q| ď K | ˆ f p a q ´ ˆ f p b q| .( ) If | L x p a ´ b q| ď ξ ¨ } a ´ b } , then | L x p a ´ b q| ď ? K ´ ¨ } a n ´ ´ b n ´ } , ROJECTION INEQUALITIES FOR ANTICHAINS 19 where a n ´ “ p a , . . . , a n ´ q and b n ´ is defined analogously. For the proof of ( ) weobserve that, similarly as before, Lemma 3.12 yields | ˆ f A p a q ´ ˆ f A p b q| ( iv ) ě | L a p a ´ b q| ´ η } a ´ b }ě | L x p a ´ b q| ´ p} L x ´ L a } ` η q ¨ } a ´ b } ( ii ) ě | L x p a ´ b q| ´ η } a ´ b } . Moreover, the definitions of ξ and η imply2 η } a ´ b } “ K ´ K ¨ ξ ¨ } a ´ b } ď K ´ K ¨ | L x p a ´ b q| , so that altogether we arrive at the desired estimate | ˆ f p a q ´ ˆ f p b q| ě K ¨ | L x p a ´ b q| . Proceeding with ( ) we set c i “ D i ˆ f A p x q for every i P r n ´ s and c “ p c , . . . , c n ´ q .Thus λ “ } c n ´ } , λ “ | c n ´ | , and L x p a ´ b q “ c ¨ p a ´ b q . Owing to the Cauchy-Schwarzinequality, the triangle inequality, and the assumption of ( ) we have λ ¨ | a n ´ ´ b n ´ | ´ λ ¨ } a n ´ ´ b n ´ } “ | c n ´ p a n ´ ´ b n ´ q| ´ } c n ´ }} a n ´ ´ b n ´ }ď | c n ´ p a n ´ ´ b n ´ q| ´ | c n ´ ¨ p a n ´ ´ b n ´ q|ď | c ¨ p a ´ b q|ď ξ ¨ } a ´ b }ď ξ ¨ | a n ´ ´ b n ´ | ` ξ ¨ } a n ´ ´ b n ´ } . Since ξ ď λ , this leads to λ ¨ | a n ´ ´ b n ´ | ď p λ ` λ q ¨ } a n ´ ´ b n ´ } , wherefore } a ´ b } ď } a n ´ ´ b n ´ } ` | a n ´ ´ b n ´ | ď p λ ` λ q λ ¨ } a n ´ ´ b n ´ } . Hence we have indeed | L x p a ´ b q| ď ξ ¨ } a ´ b } ď ? K ´ ¨ } a n ´ ´ b n ´ } , which concludes the proof. (cid:3) Given a measurable set S Ď R d , we shall say that a family V of open d -dimensionalcubes forms a Vitali covering of S if for every x P S and ε ą Q P V with x P Q and diam p Q q ă ε . Recall that by Vitali’s Covering Theorem (see e.g. [8, p.164]) in such a situation there is a countable subset U Ď V such that the members of U aremutually disjoint and S (cid:114) Ť U is null with respect to the Lebesgue measure. Proof of Lemma 3.3.
By Lemma 3.15 the collection of nice cubes forms a Vitali coveringof C . Therefore Vitali’s Covering Theorem yields countably many mutually disjoint nicecubes that cover C except for a null set. In view of the compactness of C and Lemma 3.12( i )this leads to finitely many mutually disjoint nice cubes, say Q p q , . . . , Q p N q , such that theBorel set B “ ď k Pr N s ` C X Q p k q ˘ satisfies H n ´ p B q ą ´ δ . It remains to show H n ´ p A q ď n ÿ i “ H n ´ ` π i p A q ˘ ` δ , (3.15)where A “ A X π ´ n p B q .Setting A p k q “ A X π ´ n ` C X Q p k q ˘ for every k P r N s , we infer H n ´ p A p k q q ď K p n ´ q ¨ n ÿ i “ H n ´ ` π i p A p k q q ˘ (3.16)from Lemma 3.14.Next we observe that due to A “ ď k Pr N s A p k q a repeated application of Lemma 3.4 reveals N ÿ k “ H n ´ ` π i p A p k q q ˘ “ H n ´ ` π i p A q ˘ for every i P r n s . Therefore, by summing (3.16) over k we obtain H n ´ p A q ď N ÿ k “ H n ´ ` A p k q ˘ ď K p n ´ q ¨ n ÿ i “ H n ´ ` π i p A q ˘ ď n ÿ i “ H n ´ ` π i p A q ˘ ` n ` K p n ´ q ´ ˘ and our choice of K in (3.13) leads to the desired estimate (3.15). (cid:3) ROJECTION INEQUALITIES FOR ANTICHAINS 21 § Proof of the main Theorem
Let n ě n -dimensional weak antichain with arbitrary “accuracy”by an antichain.It will be convenient to write S p x q “ ř ni “ x i for x P R n . Moreover, for every ε P ` , n ˘ we let f ε : R n Ñ R n denote the linear transformation p x , . . . , x n q ÞÝÑ ` x ´ εS p x q , . . . , x n ´ εS p x q ˘ , and set L ε “ ? ´ nε . (4.1)One checks easily that S p f ε p x qq “ p ´ nε q S p x q (4.2)and, consequently, f ε is invertible. Let us also note that f ε maps r , s n into r´ , s n . Fact 4.1.
Let ε P ` , n ˘ . ( i ) The inverse f ´ ε is L ε -Lipschitz. ( ii ) If A Ď r , s n is a weak antichain, then f ε p A q is an antichain.Proof. For part ( i ) it is enough to verify } f ε p x q ´ f ε p y q} ě p ´ nε q} x ´ y } for any two points x , y P R n . In terms of a “ x ´ y this rewrites as n ÿ i “ ` a i ´ εS p a q ˘ ě p ´ nε q n ÿ i “ a i , i.e., 2 nε n ÿ i “ a i ě p ε ´ nε q S p a q . This follows from the fact that the Cauchy-Schwarz inequality yields the even strongerestimate n ¨ n ÿ i “ a i ě S p a q . Now assume that contrary to ( ii ) we have a weak antichain A Ď r , s n and two distinctpoints f ε p x q , f ε p y q P f ε p A q with f ε p x q ď f ε p y q . Using S p f ε p x qq ă S p f ε p y qq and (4.2) weobtain S p x q ă S p y q . So for every i P r n s the assumption x i ´ εS p x q ď y i ´ εS p y q yields x i ď y i ` ε p S p x q ´ S p y qq ă y i . But x * y contradicts A being a weak antichain. (cid:3) Later it will be useful to know that in the situation of Fact 4.1( ii ) the projectioninequality (as in Proposition 3.1) applies to f ε p A q Ď r´ , s n . This is because the homothetyfrom r´ , s n onto r , s n sends f ε p A q Ď r´ , s n onto an antichain in r , s n and the H n ´ -measure gets rescaled by a factor of 2 n ´ under this map. Proof of Theorem 1.3.
We divide the argument into three steps.
Part I.
Suppose first that A is a compact set. Let ε P ` , n ˘ be arbitrary and recall thatby Fact 4.1( ii ) the projection inequality applies to f ε p A q . In combination with Lemma 3.11and Fact 4.1( i ) we obtain H n ´ p A q ď L n ´ ε ¨ H n ´ ` f ε p A q ˘ ď L n ´ ε ¨ n ÿ i “ H n ´ ` π i p f ε p A qq ˘ . Therefore it suffices to prove for every i P r n s thatlim inf ε Ñ H n ´ ` π i p f ε p A qq ˘ ď H n ´ ` π i p A q ˘ . (4.3)Fix i P r n s . Since } π i p f ε p a qq ´ π i p a q} ď n ε holds for all a P A and ε ą
0, we have π i ` f ε p A q ˘ Ď π i p A q p n ε q , where the notation is as in (3.6). Thus a complementary variant of Fact 3.5 yieldslim inf ε Ñ H n ´ ` π i p f ε p A qq ˘ ď lim inf ε Ñ H n ´ ` π i p A q p n ε q ˘ ď H n ´ ´č ε ą π i p A q p n ε q ¯ . As the compactness of A implies Ş ε ą π i p A q p n ε q “ π i p A q , we thereby arrive at (4.3). Part II.
Next we treat the more general case that A is H n ´ -measurable. For everycompact K Ď A X p , q n the result of the first part entails H n ´ p K q ď n ÿ i “ H n ´ ` π i p K q ˘ ď n ÿ i “ H n ´ ` π i p A q ˘ . (4.4)Since H n ´ p A q is finite by Corollary 3.7, the measurability of A implies (see e.g. [11,Theorem 1.7 and Theorem 1.8(ii)]) that H n ´ p A q “ sup (cid:32) H n ´ p K q : K is a compact subset of A ( . So the desired result follows from (4.4).
Part III.
Assume finally that A is an arbitrary weak antichain. We claim that theclosure ¯ A of A is likewise a weak antichain. Otherwise there existed two distinct points ROJECTION INEQUALITIES FOR ANTICHAINS 23 a , b P ¯ A such that a * b . Observe that there are sufficiently small neighbourhoods U p a q and U p b q of a and b respectively, such that c * d holds for all c P U p a q and all d P U p b q .Since U p a q and U p b q necessarily contain points of A , we get a contradiction to the factthat A is a weak antichain. This proves that ¯ A is indeed a weak antichain.Now by the Borel regularity of the Hausdorff measure, there exists for every i P r n s a Borel set B i Ě π i p A q such that H n ´ p π i p A qq “ H n ´ p B i q . Applying the result of theprevious step to the measurable weak antichain A ‹ “ ¯ A X n č i “ π ´ i p B i q , we obtain H n ´ p A q ď H n ´ p A ‹ q ď n ÿ i “ H n ´ ` π i p A ‹ q ˘ ď n ÿ i “ H n ´ p B i q “ n ÿ i “ H n ´ ` π i p A q ˘ , as required. (cid:3) § Concluding remarks
The discrete case.
For n ě A “ ∅ . This gives rise to the following question. Problem 5.1.
Determine an optimal lower bound g p n, m q on the gap ř ni “ | π i p A q| ´ | A | as A varies over weak antichains in Z n of size m . Let us mention that a slightly improved version of the argument presented in Section 2yields g p n, m q ě n ´ m, n ě
1. This can be easily seen using | A i | ă | π i p A q| if A i “ ∅ and π i p A i ´ q ‰ π i p A i q and consequently | π i p A i q| ă | π i p A q| if A i ‰ ∅ and i ě Supremum vs. Maximum.
In connection with Corollary 1.4 one may wonderfor which values of n the supremum is attained. For instance for n “ r , s has H -measure 1 and a more sophisticated construction mentionedbelow shows that for n “ Conjecture 5.2.
For every n ě r , s n whose p n ´ q -dimensionalHausdorff measure equals n . The aforementioned planar example exploits the known fact (see [13, p. 810]) that the1-dimensional Hausdorff measure of the graph of a decreasing function f : r , s ÝÑ r , s is at most 2 and that this bound is attained by singular functions (i.e., strictly decreasingfunctions whose derivative equals zero almost everywhere). It remains to observe that thegraph of a singular function is an antichain in r , s .5.3. Skewed projections of weak antichains.
So far we focused on inequalities for theorthogonal projections of weak antichains, but we believe that, actually, more generalstatements hold. In order to be more precise, we need some additional notation.Given A Ď r , s n and i P r n s , we set A i “ A X (cid:32) p x , . . . , x n q P r , s n : x i “ min t x , . . . , x n u ( . Moreover, let for i P r n s the skewed projections ∆ i : A i ÝÑ r , s n ´ be defined by p x , . . . , x n q ÞÝÑ p x ´ x i , . . . , x i ´ ´ x i , x i ` ´ x i , . . . , x n ´ x i q . Notice that the skewed projections restricted to a weak antichain are injective.
Conjecture 5.3. If A Ď r , s n is a weak antichain, then H n ´ p A q ď n ÿ i “ H n ´ ` ∆ i p A i q ˘ . Let us note that, if true, this would furnish a different proof of H n ´ p A q ď n . As a finalresult, we verify the validity of this conjecture when n “ Theorem 5.4.
Conjecture 5.3 is true when n “ .Proof. Since A is a weak antichain, it follows that ∆ is a bijection from A i onto its image.Given two numbers a, b P ∆ p A q with a ă b their inverse images under ∆ are of the form∆ ´ p a q “ p x, y q and ∆ ´ p b q “ p x ´ δ, y ` ε q for some x, y P r , s and δ, ε ě
0. Owing to } ∆ ´ p a q ´ ∆ ´ p b q} “ ? δ ` ε ď δ ` ε “ | a ´ b | the map ∆ ´ is Lipschitz with constant 1 and Lemma 3.11 yields H p A q ď H ` ∆ p A q ˘ . Applying the same reasoning to ∆ : A ÝÑ r , s we infer H p A q ď H p ∆ ` A q ˘ . From these two inequalities we conclude H p A q ď H ` ∆ p A q ˘ ` H ` ∆ p A q ˘ , as required. (cid:3) ROJECTION INEQUALITIES FOR ANTICHAINS 25
References [1] I. Anderson,
Combinatorics of finite sets , Dover Publications, Inc., Mineola, NY, 2002. Correctedreprint of the 1989 edition. MR1902962 Ò Fractals in probability and analysis , Cambridge Studies in AdvancedMathematics, vol. 162, Cambridge University Press, Cambridge, 2017. MR3616046 Ò Measure theory. Vol. I, II , Springer-Verlag, Berlin, 2007. MR2267655 Ò Measure graphs , J. London Math. Soc. (2) (1980), no. 3, 401–412, DOI 10.1112/jlms/s2-21.3.401. MR577716 Ò Representation of systems of measurable sets , Math. Proc.Cambridge Philos. Soc. (1975), no. 2, 323–325, DOI 10.1017/S0305004100051756. MR0379781 Ò On the set of divisors of a number , NieuwArch. Wisk. (1951), no. 2, 191–193. MR0043115 Ò Continuity and differentiability properties of monotone real functionsof several real variables , Math. Programming Stud. (1987), 1–16. Nonlinear analysis and optimization(Louvain-la-Neuve, 1983). MR874128 Ò Measure theory , 2nd ed., Birkhäuser Advanced Texts: Basler Lehrbücher. [BirkhäuserAdvanced Texts: Basel Textbooks], Birkhäuser/Springer, New York, 2013. MR3098996 Ò A continuous version of a Sperner-type theorem , Elektron. Informationsverarb. Kybernet. (1986), no. 1, 45–50 (English, with German and Russian summaries). MR825867 Ò
1, 1[10] ,
Sperner theory , Encyclopedia of Mathematics and its Applications, vol. 65, CambridgeUniversity Press, Cambridge, 1997. MR1429390 Ò Measure theory and fine properties of functions , Studies in AdvancedMathematics, CRC Press, Boca Raton, FL, 1992. MR1158660 Ò
1, 3.4, 4[12] K. Falconer,
Fractal geometry , John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundationsand applications. MR1102677 Ò
1, 3.4[13] J. Foran,
The length of the graph of a one to one function from r , s to r , s , Real Anal. Exchange (1999/00), no. 2, 809–816. MR1778534 Ò
1, 5.2[14] P. Frankl and R. M. Wilson,
The Erdős-Ko-Rado theorem for vector spaces , J. Combin. Theory Ser. A (1986), no. 2, 228–236, DOI 10.1016/0097-3165(86)90063-4. MR867648 Ò Continuous versions of some extremal hypergraph problems , Combinatorics (Proc.Fifth Hungarian Colloq., Keszthely, 1976), Colloq. Math. Soc. János Bolyai, vol. 18, North-Holland,Amsterdam-New York, 1978, pp. 653–678. MR519301 Ò Continuous versions of some extremal hypergraph problems. II , Acta Math. Acad. Sci. Hungar. (1980), no. 1-2, 67–77, DOI 10.1007/BF01896826. MR588882 Ò A continuous analogue of Sperner’s theorem , Comm. Pure Appl. Math. (1997), no. 3, 205–223, DOI 10.1002/(SICI)1097-0312(199703)50:3<205::AID-CPA1>3.0.CO;2-F.MR1431808 Ò Geometry of sets and measures in Euclidean spaces , Cambridge Studies in AdvancedMathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability.MR1333890 Ò [19] R. Schneider, Convex bodies: the Brunn-Minkowski theory , Second expanded edition, Encyclopedia ofMathematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014. MR3155183 Ò Ein Satz über Untermengen einer endlichen Menge , Math. Z. (1928), no. 1, 544–548,DOI 10.1007/BF01171114 (German). MR1544925 Ò Universität Rostock, Institut für Mathematik, 18051 Rostock, Germany
E-mail address : [email protected] Department of Mathematics and Applied Mathematics, University of Crete, 70013Heraklion, Greece
E-mail address : [email protected] Institute of Mathematics, Czech Academy of Sciences, Žitna 25, Praha 1, Czech Republic
E-mail address : [email protected] Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany
E-mail address ::