A Remark on discrete Brunn-Minkowski type inequalities via transportation of measure
aa r X i v : . [ m a t h . M G ] A ug A REMARK ON DISCRETE BRUNN-MINKOWSKI TYPEINEQUALITIES VIA TRANSPORTATION OF MEASURE
BOAZ A. SLOMKA
Abstract.
We give an alternative proof for discrete Brunn-Minkowski type inequalities,recently obtained by Halikias, Klartag and the author. This proof also implies strongerweighted versions of these inequalities. Our approach generalizes the ideas of Gozlan,Roberto, Samson and Tetali from the theory of optimal transportation and provides newdisplacement convexity of entropy type inequalities for the lattice point enumerator. Introduction
In the recent years, there has been a growing interest in discrete versions of inequalitiesfrom the realms of continuous functions and distributions (see e.g., [3, 4, 5, 7, 8, 12, 13, 14]).The motivation for this note was to give another proof for Brunn-Minkowski type inequalitiesfrom [6] by extending ideas from [4]. As a result, we obtain slightly stronger inequalities.1.1.
Discrete Brunn-Minkowski inequalities.
We say that an operation T : Z n × Z n → Z n admits a Brunn-Minkowski inequality if for all functions f, g, h, k : Z n → [0 , ∞ ) satisfyingthat(1) f ( x ) g ( y ) ≤ h ( T ( x, y )) k ( x + y − T ( x, y )) ∀ x, y ∈ Z n , it follows that(2) (cid:16) X x ∈ Z n f ( x ) (cid:17)(cid:16) X x ∈ Z n g ( x ) (cid:17) ≤ (cid:16) X x ∈ Z n h ( x ) (cid:17)(cid:16) X x ∈ Z n k ( x ) (cid:17) . One example for such an operation is T ( x, y ) = x ∧ y = (min( x , y ) , . . . , min( x n , y n ))which is due to the four functions theorem of Ahlswede and Daykin [1]. In this case, wehave x + y − x ∧ y = x ∨ y = (max( x , y ) , . . . , max( x n , y n )). Another example for such anoperation is due to the discrete Brunn-Minkowski inequality of Klartag and Lehec [8, Theorem1.4], which corresponds to T ( x, y ) = ⌊ ( x + y ) / ⌋ , where x + y − T ( x, y ) = ⌈ ( x + y ) / ⌉ , ⌊ x ⌋ = ( ⌊ x ⌋ , . . . ⌊ x n ⌋ ), and ⌈ x ⌉ = ( ⌈ x ⌉ , . . . , ⌈ x n ⌉ ). Here ⌊ r ⌋ = max { m ∈ Z ; m ≤ r } is thelower integer part of r ∈ R and ⌈ r ⌉ = −⌊− r ⌋ the upper integer part. t was Gozlan, Roberto, Samson and Tetali [4] who have first linked between the fourfunctions theorem of Ahlswede and Daykin and the discrete Brunn-Minkowksi inequality ofKlartag and Lehec. In their paper, they provided alternative proofs for these results whichare based on ideas from the theory of optimal transport.Recently, a unified elementary proof for the two aforementioned results was given in [6].This proof applies to all operations T : Z n × Z n → Z n sharing two common properties:(P1) Translation equivariance : T ( x + z, y + z ) = T ( x, y ) + z for all z ∈ Z n .(P2) Monotonicity in the sense of Knothe : there exists a decomposition of Z n into a directsum of groups Z n = G × · · · × G k such that for each i ∈ { , . . . , k } :(i) T i : ( G × · · · × G i ) × ( G × · · · × G i ) → G i where T = ( T , . . . , T k ). Inother words, T i ( x, y ) depends only on the first i coordinates of its arguments x, y ∈ G × · · · × G k , so that T is triangular.(ii) There exists a total additive ordering (cid:22) i on G i such that T ( a,b ) i : G i × G i → G i defined by T ( a,b ) i ( x, y ) = T i (cid:0) ( a, x ) , ( b, y ) (cid:1) for a, b ∈ G × · · · × G i − satisfies x (cid:22) i x , y (cid:22) i y = ⇒ T ( a,b ) i ( x , y ) (cid:22) i T ( a,b ) i ( x , y )for all a, b ∈ G × · · · × G i − and x , x , y , y ∈ G i .Recall that a total ordering (cid:22) on an abelian group G is a binary relation which is reflexive,anti-symmetric and transitive, such that for any distinct x, y , either x (cid:22) y or else y (cid:22) x . Anordering (cid:22) is additive if for all x, y, z , we have x (cid:22) y = ⇒ x + z (cid:22) y + z .Examples for additive, total orderings on Z n (or on R n ) are the standard lexicographicorder relation and invertible linear images thereof. The requirement of existence of a totaladditive ordering on a finitely-generated abelian group G , forces G to be isomorphic to Z ℓ forsome ℓ . Note that properties (P1) and (P2) are closed under cartesian products of operations. Theorem 1.1 ([6, Theorem 1.3]) . Every translation equivariant operation T : Z n × Z n → Z n which is monotone in the sense of Knothe admits a Brunn-Minkowski inequality. We remark that Knothe [9] used maps satisfying a condition similar to (P2) in his proofof the Brunn-Minkowski inequality.In addition to the four functions theorem and the Brunn-Minkowski inequality of Klartagand Lehec, Theorem 1.1 implies various other inequalities, some of which are related to worksof Ollivier and Villani [14], Iglesias, Yepes Nicol´as and Zvavitch [7], and Cordero-Erausquinand Maurey [2]. For more details see [6]. ur first main theorem is the following: Theorem 1.2.
Let α, β, γ, δ > such that max { α, β } ≤ min { γ, δ } . Let T : Z n × Z n → Z n satisfy properties (P1) and (P2) and suppose that f, g, h, k : Z n → [0 , ∞ ) satisfy f α ( x ) g β ( y ) ≤ h γ ( T ( x, y )) k δ ( x + y − T ( x, y )) ∀ x, y ∈ Z n . Then (cid:16) X x ∈ Z n f ( x ) (cid:17) α (cid:16) X x ∈ Z n g ( x ) (cid:17) β ≤ (cid:16) X x ∈ Z n h ( x ) (cid:17) γ (cid:16) X x ∈ Z n k ( x ) (cid:17) δ . Note that if an operation T satisfies properties (P1) and (P2), then so does the operation x + y − T ( x, y ). In the sequel, we shall denote such pairs of “complementing” operations by T − and T + .1.2. A discrete displacement convexity of entropy type result.
Our approach is in-spired by the work of Gozlan, Roberto, Samson and Tetali [4] who proved the following resultfor the counting measure m on Z , T − ( x, y ) = ⌊ ( x + y ) / ⌋ and T + ( x, y ) = ⌈ ( x + y ) / ⌉ : Theorem 1.3 ([4, Theorem 8]) . Suppose that µ , µ are finitely supported probability mea-sures on Z . Then (3) H ( µ | m ) + H ( µ | m ) ≥ H ( µ − | m ) + H ( µ + | m ) where H ( µ | ν ) = P x ∈ Z µ ( x ) log( µ ( x ) ν ( x ) ) is the relative entropy of µ with respect to ν , and µ ± isthe push forward of the monotone coupling π between µ and µ by T ± . Denote the counting measure on Z n by m n . The relative entropy of a probability measure µ on Z n with respect to m n is given by H ( µ | m n ) = P x ∈ Z n µ ( x ) log( µ ( x )).We prove the following: Theorem 1.4.
Let α, β, γ, δ > such that max { α, β } ≤ min { γ.δ } . Let T ± : Z n × Z n → Z n be complementing operations satisfying properties (P1) and (P2). Suppose that µ and ν arefinitely supported probability measures on Z n . Then there exists a coupling π between µ and ν such that, denoting by κ ± = π ◦ T ±− the push forward of π by T ± , we have (4) αH ( µ | m n ) + βH ( ν | m n ) ≥ γH ( κ − | m n ) + δH ( κ + | m n ) . The coupling for which Eq. (4) holds is (perhaps not at all surprisingly) a Knothe couplingwhich is compatible to the decomposition of Z n = G × · · · × G k , given in property (P2).Theorem 1.4 is a discrete variant of the convexity of entropy property, put forward bySturm [16]. As observed in [4], Theorem 1.4 implies Theorem 1.2 by duality. A similar uality argument was used by Lehec in [10] to obtain reversed Brascamp-Lieb inequalities.For additional discrete results in the this spirit, see [3], [14] and references therein.Theorem 1.4 is an immediate consequence of the following extension of [4, Theorem 9](which was used to deduce Theorem 1.3 in the same manner): Theorem 1.5.
Let α, β, γ, δ > such that max { α, β } ≤ min { γ.δ } . Let T ± : Z n × Z n → Z n be complementing operations satisfying properties (P1) and (P2). Suppose that µ and ν arefinitely supported probability measures on Z n . Then there exists a coupling π between µ and ν such that, denoting by κ ± = π ◦ T ±− the push forward of π by T ± , we have P := X ( x,y ) ∈ Z n × Z n κ γ − ( T − ( x, y )) κ δ + ( T + ( x, y )) µ α ( x ) ν β ( y ) π ( x, y ) ≤ . The remaining of the paper is organized as follows. In Section 2 we prove Theorem 1.5in the case where T itself is monotone. In Section 3 we use the Knothe coupling to extendthe proof of Theorem 1.5 to the general case. Sections 4 and 5 are devoted for the proofs ofTheorems 1.3 and 1.2, respectively. Acknowledgement.
The author thanks Bo’az Klartag for fruitful conversions and for his adviceand comments. The author also thanks Shiri Artstein for her remarks on the written textand the anonymous referee of the paper [6] for suggesting to pursue this direction.2.
Monotone couplings for totally ordered groups
The purpose of this section is to prove Theorem 1.4 in the case where T itself is monotonein each of its two entries with respect to some total additive ordering (cid:22) on Z n . This resultis given below as Proposition 2.1.The core ideas of our proof of Proposition 2.1 are drawn from the proof of Theorem 1.3 in[4]. However, we also manage to simplify some of the key steps there, which is mainly thanksto the fact that we consider an abstract operation T rather than a specific one.Let G be a finitely generated group, endowed with a totally additive ordering (cid:22) . Recallthat G ≈ Z l for some l .Given a probability measure µ on G , the cumulative distribution of µ with respect to (cid:22) isdefined by F µ ( x ) = µ (( −∞ , x ]) = µ { g ∈ G ; g (cid:22) x } ∀ x ∈ G. Similarly, the generalized inverse of F µ at a point t ∈ (0 ,
1) is given by F − µ ( t ) = inf { x ∈ G ; F µ ( x ) ≥ t } . iven two finitely supported probability measures µ, ν on G and a random variable U ,uniformly distributed on (0 , µ and ν withrespect to the ordering (cid:22) by π = Law( F − µ ( U ) , F − ν ( U )) . It is not hard to check that the support of π is monotone with respect to (cid:22) × (cid:22) . That is,if ( a, b ) , ( c, d ) ∈ supp( π ) then either a (cid:22) c and b (cid:22) d or vice versa c (cid:22) a and d (cid:22) a . Indeed,suppose otherwise that a (cid:22) c and d (cid:22) b with ( a, b ) = ( c, d ) and that there exist t , t ∈ (0 , F − ( t ) , G − ( t )) = ( a, b ) and ( F − ( t ) , G − ( t )) = ( c, d ). Then, on the one hand, a (cid:22) b , implies that t > t and, on the other hand, d (cid:22) b implies that t > t , a contradiction.We remark that if ν stochastically dominates µ , or the other way around, then the coupling π is diagonal, i.e., F − µ ( U ) ≤ F − ν ( U ) with probability 1 or 0. This is a particular caseof Strassen’s theorem [15], which holds for partially ordered probability spaces. For moreinformation on this subject, see e.g., [11] and references therein. Proposition 2.1.
Let T ± : G × G → G be complementing operations satisfying properties(P1) and (P2). Let α, β, γ, δ > such that max { α, β } ≤ min { δ, γ } . Suppose that µ and ν are finitely supported probability measures on G and let π be the monotone coupling between µ and ν with respect to the ordering given in (P2). Then, denoting κ ± = π ◦ T ±− , we have αH ( µ | m n ) + βH ( ν | m n ) ≥ δH ( κ − | m n ) + γH ( κ + | m n ) . Proposition 2.1 is an immediate consequence of the following proposition. For the proof ofthis implication, see Section 4 below.
Proposition 2.2.
With the same notation as in Proposition 2.1, we have X ( x,y ) ∈ G × G κ γ − ( T − ( x, y )) κ δ + ( T + ( x, y )) µ α ( x ) ν β ( y ) π ( x, y ) ≤ . To prove Proposition 2.2, we shall need the following lemmas, in which we shall use thenotation 1 = min { g ∈ G : 0 ≺ g } . Lemma 2.3.
Suppose that ( x , y ) = ( x , y ) , x (cid:22) x and y (cid:22) y . If T ± ( x , y ) = T ± ( x , y ) then x − x + y − y = 1 and T ∓ ( x , y ) = T ∓ ( x , y ) + 1 .Proof. If y + 1 (cid:22) y and x + 1 (cid:22) x , then 1 + T ± ( x , y ) = T ± ( x + 1 , y + 1) (cid:22) T ± ( x , y ).Moreover, if T ± ( x , y ) = T ± ( x , y ), then the relation T ± ( x, y ) + T ∓ ( x, y ) = x + y impliesthat T ∓ ( x , y ) = x + y − T ± ( x , y ) = x + y − ( x + y − T ∓ ( x , y )) = 1 + T ∓ ( x , y ) . (cid:3) enote Im( T ± ) = { T ± ( x, y ) ; ( x, y ) ∈ supp( π ) } . For a ∈ G denote S ± ( a ) = { ( x, y ) ∈ supp( π ) ; T ± ( x, y ) = a } where S ± ( a ) = ∅ when a Im( T ± ). Lemma 2.4.
For every a ∈ G , we have Card( S ± ( a )) ∈ { , , } . If Card( S ± ( a )) = 2 then S ± ( a ) = { ( x , y ) , ( x , y ) } where ( x , y ) = ( x , y + 1) or ( x , y ) = ( x + 1 , y ) .Proof. Let S = S − (the proof for S = S + is done verbatim). Suppose Card( S ( a )) >
1. Let x be the minimal first coordinate of the elements of S ( a ) and y be the minimal secondcoordinate of the elements of S ( a ) having x as first coordinate. If T − ( x , y ) = T − ( x , y )for some other ( x , y ) ∈ S − ( a ) then, by the definition of ( x , y ) and the monotonicity of thesupport of π , we have x (cid:22) x and y (cid:22) y . By Lemma 2.3, it follows x − x + y − y = 1which, in turn, implies that x = x and y = y + 1 or x = x + 1 and y = y . By themonotonicity of the support of π , these cases exclude each other, thus Card( S ( a )) = 2. (cid:3) For the next lemma we need the following definition: given ( x, y ) ∈ supp( π ), denote S ± ( x, y ) = S ± ( T ± ( x, y )) for brevity. We say that S − ( x, y ) and S + ( x, y ) are aligned if for each( x ′ , y ′ ) ∈ S ± ( x, y ) there exists ( x ′′ , y ′′ ) ∈ S ∓ ( x, y ) such that( x ′ , y ′ ) ∈ { ( x ′′ , y ′′ ) , ( x ′′ + 1 , y ′′ + 1) , ( x ′′ − , y ′′ − } . By Lemma 2.4, if S − ( x, y ) and S + ( x, y ) are aligned then Card( S − ( x, y ))=Card( S + ( x, y )).Also note that if S − ( x, y ) and S + ( x, y ) are not aligned then either Card( S − ( x, y )) = 2 orCard( S + ( x, y )) = 2. Lemma 2.5. If Card( S − ( x, y )) = Card( S + ( x, y )) = 2 then S − ( x, y ) and S + ( x, y ) are aligned.Proof. Let ( x , y ) be a point whose first coordinate is minimal such that S − ( x, y ) and S + ( x, y )are not aligned and Card( S − ( x , y )) = Card( S + ( x , y )) = 2.Denote a = T − ( x , y ) and a ′ = T + ( x , y ). By interchanging the roles of S + and S − , andthe first and second coordinates if needed, we may assume without loss of generality that S − ( x , y ) = S − ( a ) = { ( x , y ) , ( x + 1 , y ) } . Since S − ( x , y ) and S + ( x , y ) are not aligned and ( x , y + 1) is excluded from supp( π ), S + ( x , y ) = S + ( a ′ ) = { ( x − , y ) , ( x , y ) } . By the alignment of S ± ( x − , y ) (due to the minimality of x ) and Lemma 2.4, we have S − ( x − , y ) = S − ( a −
1) = { ( x − , y − , ( x − , y ) } . gain, since S − ( x − , y −
1) and S + ( x − , y ) are aligned, it follows by Lemma 2.4 that S + ( x − , y −
1) = S + ( a ′ −
1) = { ( x − , y − , ( x − , y − } . Continuing this process indefinitely, we obtain a contradiction to the finiteness of supp( π ). (cid:3) An immediate consequence of Lemma 2.5 is the following:
Lemma 2.6.
Let a ∈ G and assume that ( x , y ) ∈ S − ( a ) . Denote a ′ = T + ( x , y ) . Supposethat Card( S − ( a ′ )) = 2 and that S − ( a ) = { ( x , y ) , ( x , y ) } , with x (cid:22) x (cid:22) x + 1 and y (cid:22) y (cid:22) y + 1 . Then S + ( a ′ ) = { ( x − , y − , ( x , y ) } . Proof of Proposition 2.2.
We shall actually show that for all ( x, y ) ∈ supp( π ) we have(5) κ γ − ( T − ( x, y )) κ δ + ( T + ( x, y )) µ α ( x ) ν β ( y ) ≤ . This would clearly imply the desired result.Fix ( x , y ) ∈ supp( π ) and denote a = T − ( x , y ) and a ′ = T + ( x , y ). We show that (5)holds for ( x , y ) by considering two cases: Case 1:
Either Card( S − ( a )) = 1 or Card( S + ( a ′ )) = 1. By switching the roles of S − ( a ) and S + ( a ′ ), we may assume without loss of generality that S − ( a ) = { ( x , y ) } . By Lemma 2.3,either S + ( a ′ ) ⊆ { ( x , y ) , ( x ± , y ) } or S + ( a ′ ) ⊆ { ( x , y ) , ( x , y ± } . Therefore, either κ + ( a ′ ) ≤ π ( x , y ) + π ( x , y ± ≤ µ ( x ) or κ + ( a ′ ) ≤ π ( x , y ) + π ( x ± , y ) ≤ ν ( y ) . Since κ − ( a ) = π ( x , y ) and π ( x , y ) ≤ min( µ ( x ) , ν ( y )), the desired inequality (5) follows. Case 2:
Card( S − ( a )) = Card( S + ( a ′ )) = 2. As in the previous case, by the symmetrybetween S − ( a ) and S + ( a ′ ) , we may assume without loss of generality that S − ( a ) = { ( x , y ) , ( x , y + 1) } . By Lemma 2.6, we have S + ( a ′ ) = { ( x − , y ) , ( x , y ) } . Thus κ − ( a ) = π ( x , y ) + π ( x , y + 1) ≤ µ ( x ) , and κ + ( a ′ ) = π ( x , y ) + π ( x − , y ) ≤ ν ( y ) . which establishes (5) and completes the proof. (cid:3) . Knothe couplings between measures on Z n Denote by Z n = ( G i , (cid:22) i ) k the decomposition of Z n into a direct sum of groups, G , . . . , G k ,each of which equipped with a total additive ordering (cid:22) i . For each i , and any two finitelysupported probability measures µ and ν on G i , let π i be the monotone coupling between µ and ν , defined in the Section 2.Next, we construct the Knothe coupling π between two finitely supported measures µ and ν on Z n with respect to this decomposition.To that end, the following notation shall be useful. For ( x , . . . , x k ) ∈ ( G , . . . , G k ) andeach i ∈ { , . . . , k } denote by x i the sub-vector ( x , . . . , x i ) ∈ ( G , . . . , G i ). Consider thedisintegration formula for a measure κ on Z n with respect to the given decomposition: κ ( x , . . . , x k ) = κ ( x ) κ ( x | x ) . . . κ k ( x k | x k − )where κ is the marginal of κ onto G , κ ( · | x ) is the marginal of κ ( · | x ) onto G and etc.The Knothe coupling between µ and ν with respect to this decomposition is defined by π ( x, y ) = π ( x , y ) π ( x , y | x , y ) . . . π k ( x k , y k | x k − , y k − )where π i ( · , · | x i − , y i − ) is the monotone coupling between µ i ( · | x i − ) and ν i ( · | y i − ). Proof of Theorem 1.5.
Recall that since T ± : Z n × Z n → Z n are monotone in the sense ofKnothe with respect to the decomposition ( G i , (cid:22) i ) k , we have T ± ( x, y ) = ( T ± ( x , y ) , T ± ( x , x , y , y ) . . . , T k ± ( x k − , y k − ))for all x = ( x , . . . , x k ) , y = ( y , . . . , y k ) ∈ ( G , . . . , G k ). Moreover, for each i ∈ { , . . . , k } ,the operations ( T i ± ) ( x i − ,y i − ) : G i × G i → G i , defined by( T i ± ) ( x i − ,y i − ) ( x i , y i ) = T i ± ( x i , y i )for all x i − , y i − ∈ G × · · · × G i − , are increasing in each of their two entries.Let π be the Knothe coupling between µ and ν with respect to the same decomposition Z n = ( G i , (cid:22) i ) k , and recall that κ ± = π ◦ T ±− . Then P = X ( x,y ) ∈ Z n × Z n κ γ − ( T − ( x, y )) κ δ + ( T + ( x, y )) µ α ( x ) ν β ( y ) π ( x, y )= X ( x ,y ) ∈ G X ( x ,y ) ∈ G · · · X ( x k ,y k ) ∈ G k κ γ − ( T − ( x, y )) κ δ + ( T + ( x, y )) µ α ( x ) ν β ( y ) π ( x, y ) . sing the disintegration of κ ± and the fact that T ± are triangular with respect to the givendecomposition of Z n , we have κ ± ( T ± ( x, y )) = κ ± ( m ± ( x , y )) κ ± ( m ± ( x , y ) | x , y ) . . . κ k ± ( m k ± ( x k , y k ) | x k − , y k − )where, for brevity, the expression m i ± ( x i , y i ) within κ i ( m i ± ( x i , y i ) | x i − , y i − ) is understoodas ( m i ± ) ( x i − ,y i − ) ( x i , y i ). Combined with the disintegration of µ , ν , and π with respect tothe given decomposition of Z n , we obtain that P = X ( x ,y ) ∈ G × G A ( x , x ) X ( x ,y ) ∈ G × G A ( x ,y )2 ( x , y ) · · · X ( x k ,y k ) ∈ G k × G k A ( x k − ,y k − ) k ( x k , y k )where A ( x i − ,y i − ) i ( x i , y i ) is given by (cid:16) κ i − ( m i − ( x i , y i ) | x i − , y i − ) (cid:17) γ (cid:16) κ i + ( m i + ( x i , y i ) | x i − , y i − ) (cid:17) δ (cid:16) µ i ( x i | x i − ) (cid:17) α (cid:16) ν i ( y i | y i − ) (cid:17) β π i ( x i , y i | x i − , y i − ) . Finally, we apply Proposition 2.1 iteratively to each sum separately to obtain that X ( x i ,y i ) ∈ G i A ( x i − ,y i − ) i ( x i , y i ) ≤ i ∈ { , . . . , k } and x i − , y i − ∈ G × · · · × G i − . This completes the proof. (cid:3) Proof of Theorem 1.4
Let π be the Knothe coupling between µ and ν , given in Theorem 1.5. By Jensen’sinequality, applied to the logarithm function, Theorem 1.5 implies that H := X ( x,y ) ∈ Z n × Z n log (cid:18) κ γ − ( T − ( x, y )) κ δ + ( T + ( x, y )) µ α ( x ) ν β ( y ) (cid:19) π ( x, y ) ≤ π , κ − and κ + we have H = γ X z ∈ Z n log( κ − ( z )) κ − ( z ) + δ X z ∈ Z n log( κ + ( z )) κ + ( z ) − α X z ∈ Z n log( µ ( z )) µ ( z ) − β X z ∈ Z n log( ν ( z )) ν ( z )= γH ( κ − | m n ) + δH ( κ + | m n ) − αH ( µ | m n ) − βH ( ν | m n ) ≤ . Proof of Theorem 1.2
As in [4], we use the log-Laplace transform of any bounded function ϕ :(6) log Z e ϕ dm n = sup ν { Z ϕ dν − H ( ν | m n ) } . et f, g, h, k satisfy(7) f α ( x ) g β ( y ) ≤ h γ ( T − ( x, y )) k δ ( T + ( x, y )) ∀ x, y ∈ Z n . If either h or k is not bounded from above then the statement holds trivially. Otherwise,it follows from (7) that f, g, h, k are all bounded from above. Given ε > f ε = max( ε, f ( x )), one may check that the above inequality is equivalent to α log f ε ( x ) + β log g ε ( y ) ≤ γ log h ε ( T − ( x, y )) + δ log k ε ( T + ( x, y )) . Integrating this inequality with respect to the Knothe coupling π between finitely supportedprobability measures µ, ν on Z n , as given in Theorem 1.4, we have α Z log f ε dµ + β Z log g ε dν ≤ γ Z log( h ε ◦ T − ) dπ + δ Z log( k ε ◦ T + ) dπ = Z γ log h ε dκ − + δ Z log k ε dκ + , where κ ± = π ◦ T ±− . Applying Theorem 1.4 and (6) we thus get α (cid:16) Z log f ε dµ − H ( µ | m n ) (cid:17) + β (cid:16) Z log g ε dν − H ( ν | m n ) (cid:17) ≤ γ (cid:16) Z log h ε dκ − − H ( κ − | m n ) (cid:17) + δ (cid:16) Z log k ε dκ + − H ( κ + | m n ) (cid:17) ≤ γ log Z h ε dm n + δ log Z k ε dm n . Optimizing over all µ and ν , we get α log Z f ε dm n + β log Z g ε dm n ≤ γ log Z h ε dm n + δ log Z k ε dm n . We conclude the proof by taking ε → References [1] R. Ahlswede and D. E. Daykin,
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Department of Mathematics, the Open University of Israel, Ra’anana 4353701 Israel
E-mail address : [email protected]@openu.ac.il