Blaschke-Santaló inequalities for Minkowski and Asplund endomorphisms
aa r X i v : . [ m a t h . M G ] J a n Blaschke–Santal´o Inequalities forMinkowski and Asplund Endomorphisms
Georg C. Hofst¨atter and Franz E. Schuster
Abstract.
It is shown that each monotone Minkowski endomorphism ofconvex bodies gives rise to an isoperimetric inequality which directly impliesthe classical Urysohn inequality. Among this large family of new inequalities,the only affine invariant one – the Blaschke–Santal´o inequality – turns out tobe the strongest one. A further extension of these inequalities to merelyweakly monotone Minkowski endomorphisms is proven to be impossible.Moreover, for functional analogues of monotone Minkowski endomorphisms,a family of analytic inequalities for log-concave functions is established whichgeneralizes the functional Blaschke–Santal´o inequality.
1. Introduction
One of the most widely known and fundamental affine isoperimetric inequalitiesis the
Blaschke–Santal´o inequality , roughly stating that the volume product of polarreciprocal convex bodies is maximized by ellipsoids. More precisely, let K ⊆ R n be a convex body (that is, a compact, convex set) with non-empty interior andrecall that K z = { x ∈ R n : x · y ≤ y ∈ K − z } is the polar body of K with respect to z ∈ int K . Denoting by | A | the volume of a Borel set A ⊆ R n ,the Santal´o point can be defined as the unique point s = s ( K ) ∈ int K , for which | K s | = min {| K z | : z ∈ int K } . The Blaschke–Santal´o inequality then states that | K || K s | ≤ | B n | (1.1)with equality if and only if K is an ellipsoid (that is, an affine image of the Euclideanunit ball B n ). Initial proofs of (1.1) were given in the first half of the previouscentury by Blaschke for n ≤ n ≥
2, while the equalityconditions were completely settled only in 1985 by Petty. In subsequent years,simplified proofs, including the equality cases, were obtained (see, e.g., [ , ])and it remained an active focus of research due to the evolving understanding of itsimpact (see [ , , , , , ] and the references therein).Affine invariant inequalities are often more powerful than related inequalitiesthat are merely invariant under Euclidean rigid motions. This becomes particularlystriking for the Blaschke–Santal´o inequality which considerably strengthens anddirectly implies the classical Urysohn inequality (as first observed by Lutwak [ ]).The latter is the following basic relation between the mean width w ( K ) of a convexbody K ⊆ R n (see Section 2 for definition) with non-empty interior and its volume, | K | ≤ (cid:18) w ( K )2 (cid:19) n | B n | (1.2)with equality if and only if K is a ball. 1nother affine isoperimetric inequality that plays a special role in this papercoincides for origin-symmetric bodies with the Blaschke–Santal´o inequality but isin general weaker than (1.1). In order to state it, let ∆ K = ( − K + K ) denote the central symmetral of a convex body K ⊆ R n . If K has non-empty interior, then(1.1), combined with the Brunn–Minkowski inequality, implies that | K || ∆ ◦ K | ≤ | B n | (1.3)with equality if and only if K is an ellipsoid. Here, ∆ ◦ K is the polar body of ∆ K with respect to the origin. The central symmetrization ∆ has long been a useful toolin the Brunn–Minkowski theory (see, e.g., [ , Chapter 3.2] and [ , Chapter 10.1]).As a continuous operator on the space K n of convex bodies in R n endowed withthe Hausdorff metric, the importance of ∆ stems from its Minkowski additivity (that is, ∆( K + L ) = ∆ K + ∆ L for all K, L ∈ K n ) and compatibility with affinetransformations – characterizing properties, as the following result shows. Theorem (Schneider [ ]) A continuous map
Φ : K n → K n is a translationinvariant Minkowski additive map such that Φ( AK ) = A Φ K for every K ∈ K n and A ∈ GL( n ) if and only if Φ = c ∆ for some c ≥ . This theorem was a byproduct of a more general, systematic study of Minkowskiadditive operators on K n , initiated about 50 years ago by Schneider [ ]. Sincethen, and up to now, the main focus thereby has been on maps that also commutewith SO( n ) transforms (see [ , , , ]). As such maps are automaticallycompatible with translations (see, e.g., [ , Section 2.3]), they are often assumedw.l.o.g. to be translation invariant, leading to the following central definition. Definition
A continuous map
Φ : K n → K n is a Minkowski endomorphism if Φ is Minkowski additive, translation invariant, and commutes with SO( n ) transforms.The trivial Minkowski endomorphism maps every convex body to the origin.
Much of this article is motivated by the observation that (1.2) and (1.3) can becast as volume estimates for polar Minkowski endomorphisms. Another prominentsuch example was established by Lutwak [ ] for polar projection bodies of order 1, | K || Π ◦ K | ≤ | B n | (1.4)with equality if and only if K is a ball. Recalling that each K ∈ K n is determinedby its support function h ( K, u ) = max { u · x : x ∈ K } for u ∈ S n − , Π K can bedefined by h (Π K, u ) = c n w ( K | u ⊥ ), where c n ∈ R is chosen such that Π B n = B n .The natural question to what degree inequalities (1.2), (1.3), and (1.4) can beunified, was first asked by Lutwak. A partial answer was given in [ ], deducedfrom results in [ ], where (1.2) and (1.4) were identified as part of a larger familyof inequalities for a subcone of Minkowski endomorphisms which are monotone ,that is, K ⊆ L implies Φ K ⊆ Φ L for all K, L ∈ K n . For a more precise statementwe require the following classification of monotone Minkowski endomorphisms.2 heorem (Kiderlen [ ]) A map
Φ : K n → K n is a monotone Minkowski endo-morphism if and only if there exists a non-negative SO( n − invariant measure µ on S n − with center of mass at the origin such that h (Φ K, · ) = h ( K, · ) ∗ µ (1.5) for every K ∈ K n . Moreover, the measure µ is uniquely determined by Φ . The convolution of functions and measures on S n − used in (1.5) is inducedfrom SO( n ) by identifying S n − with the homogeneous space SO( n ) / SO( n −
1) (seeSection 2 for details). Note that we assume all measures to be finite Borel measures.In [ ], (1.2) and (1.4) were generalized to monotone Minkowski endomorphismsgenerated by area measures of order one of zonoids (see Section 2). As a firstmain result, we generalize these inequalities from [ ] to all monotone Minkowskiendomorphisms Φ. Throughout, we always assume that n ≥ Theorem 1
Suppose that
Φ : K n → K n is a monotone non-trivial Minkowskiendomorphism. Among K ∈ K n with non-empty interior the volume product | K || Φ ◦ K | is maximized by Euclidean balls. If Φ = c ∆ for some c > , then K is a maximizerif and only if it is an ellipsoid. Otherwise, Euclidean balls are the only maximizers. Let us emphasize that Theorem 1 not only includes inequalities (1.2), (1.3), and(1.4) as special cases, but provides an extension of the isoperimetric inequalitiesfrom [ ] from a nowhere dense set of Minkowski endomorphisms to all monotoneones. Whereas the proof of Theorem 1 does not require any results from [ ], ourapproach is very much inspired by techniques from [ ] and relies on Kiderlen’sclassification of monotone Minkowski endomorphisms.While it was long known that not all Minkowski endomorphisms are monotone,a conjecture that they are all weakly monotone (see Section 3 for details) wasdisproved by Dorrek [ ] only recently. We will show in Section 4 that Theorem 1is essentially best possible, in the sense that a further extension to all merely weaklymonotone endomorphisms is in general impossible.By Schneider’s above characterization of the map ∆, inequality (1.3) is theonly affine invariant one among the family of isoperimetric inequalities provided byTheorem 1. With our second main result, we show that all these inequalities can bededuced from the Blaschke–Santal´o inequality. In particular, among inequalities foreven Minkowski endomorphisms, (1.3) is the strongest member of the inequalitiesfrom Theorem 1. This is in contrast to the volume estimates obtained in [ ],among which (1.4) was the strongest one, since (1.3) was not included. Finally, weprove that each of the inequalities of Theorem 1 is stronger and directly implies theUrysohn inequality (1.2). 3 heorem 2 If Φ : K n → K n is a monotone Minkowski endomorphism such that Φ B n = B n and K ∈ K n has non-empty interior, then | B n | (cid:18) w ( K )2 (cid:19) − n ≤ | Φ ◦ K | ≤ | K s | . (1.6) There is equality in the left hand inequality if and only if Φ K is a Euclidean ball.Equality in the right hand inequality holds if and only if K is centrally-symmetricand Φ = ∆ or if K is a Euclidean ball. Note that the right inequality of Theorem 2 combined with the Blaschke–Santal´oinequality implies Theorem 1. In Section 4, we will therefore first prove Theorem 2and then deduce Theorem 1 as a consequence.A second focus of this article concerns the continuing effort to extend notionsand results from convex geometry to the class of log-concave functions, that is, all f : R n → [0 , ∞ ) of the form f = e − ϕ for some convex ϕ : R n → ( −∞ , ∞ ]. The mostbasic such notions are Minkowski addition and scalar multiplication which can benaturally extended as follows. For log-concave f and g and λ >
0, let( f ⋆ g )( x ) = sup x + x = x f ( x ) g ( x ) , ( λ · f )( x ) = f (cid:0) xλ (cid:1) λ . Then f ⋆g is called the
Asplund sum (or sup-convolution) of f and g (see, e.g., [ ]).While the above definitions imply that K ⋆ L = K + L and λ · K = λK for allindicators of K, L ∈ K n and λ >
0, in general it is possible that f ⋆ g attains thevalue + ∞ and, thus, is no longer log-concave. Moreover, the standard regularityassumption of upper semi-continuity of log-concave functions need not be preservedunder Asplund addition (cf. [ , p. 517]). One frequently used possibility toovercome these issues is to work with the space LC c ( R n ) of all proper log-concavefunctions which are upper semi-continuous and coercive. Here, f is called proper if it is not identically 0 and it is coercive if lim k x k→∞ f ( x ) = 0. We furthermoreendow LC c ( R n ) with the topology induced by epi-convergence (see Section 2).As a seminal inequality for log-concave functions, we first mention the celebratedPr´ekopa–Leindler inequality which is universally recognized as the functional formof the Brunn–Minkowski inequality (see, e.g., [ , Section 7.1]). A functionalversion of the Blaschke–Santal´o inequality, discovered later by Ball [ ], is of specialimportance for us. Hence, recall that for log-concave f : R n → [0 , ∞ ), the polarfunction of f can be defined (following [ ]) by f ◦ = e −L ( − log f ) , (1.7)where L denotes the classical Legendre transform (see Section 2). If f is in addition even and 0 < R R n f dx < ∞ , then Ball’s functional Blaschke–Santal´o inequality reads Z R n f ( x ) dx Z R n f ◦ ( x ) dx ≤ (2 π ) n (1.8)with equality if and only if f is a Gaussian.4he cases for equality in (1.8) were settled by Artstein-Avidan, Klartag andMilman [ ], who also established a far-reaching extension of (1.8) to not necessarilyeven functions (cf. Section 2), that has sparked a great deal of research interest inrecent years, see [ , , , , , ].As noted by Rotem [ ], the functional Blaschke–Santal´o inequality implies ananalogue of Urysohn’s inequality for log-concave functions, a result first obtainedby Klartag and Milman [ ] by other means. It can be conveniently formulatedwith the help of the support function of a log-concave f : R n → [0 , ∞ ) which is,following [ ], defined by h ( f, · ) = L ( − log f ). If additionally R R n f dx = (2 π ) n/ and γ n is the standard Gaussian measure on R n , the functional analogue of Urysohn’sinequality states that 2 n Z R n h ( f, x ) dγ n ( x ) ≥ π ) − n/ f is a translation of the standard Gaussian.Note that if f = K for some K ∈ K n , the left hand side is proportional to w ( K ).However, the sharp geometric Urysohn inequality cannot be recovered from (1.9).Recalling that SO( n ) acts naturally on LC c ( R n ), specifically, ( ϑf )( x ) = f ( ϑ − x )for ϑ ∈ SO( n ) and f ∈ LC c ( R n ), we can now introduce ‘functional Minkowskiendomorphisms’ on LC c ( R n ) as follows. Definition
A continuous map
Ψ : LC c ( R n ) → LC c ( R n ) is an Asplund endomor-phism if Ψ is Asplund additive, translation invariant, and commutes with the SO( n ) action. The trivial Asplund endomorphism maps every function to the indicator ofthe origin.
In Section 3, we discuss the problem to establish an analogue of Kiderlen’scharacterization of monotone Minkowski endomorphisms. Here, we use the latteras motivation to define the following rich class of Asplund endomorphisms whichare monotone , that is, f ≤ g implies Ψ f ≤ Ψ g for all f, g ∈ LC c ( R n ). Theorem 3
Each non-negative
SO( n − invariant measure µ on S n − with centerof mass at the origin induces a monotone Asplund endomorphism Ψ µ by h (Ψ µ f, · ) = h ( f, · ) ⊛ µ for f ∈ LC c ( R n ) . Moreover, the measure µ is uniquely determined by Ψ µ . For the definition of the convolution ⊛ of the convex function h ( f, · ) on R n with the measure µ , we refer to Section 2. Let us emphasize that Ψ µ K = Φ µ K for every K ∈ K n , where Φ µ is the monotone Minkowski endomorphism definedby (1.5). In this sense, the Asplund endomorphisms Ψ µ extend the class of allmonotone Minkowski endomorphisms to LC c ( R n ). As our next main result, weprove a functional analogue of Theorem 1 for the monotone Asplund endomorphismsdefined by Theorem 3. 5 heorem 4 Let µ be an SO( n − invariant probability measure on S n − withcenter of mass at the origin. If f ∈ LC c ( R n ) such that R R n f dx > , then Z R n f ( x ) dx Z R n (Ψ µ f ) ◦ ( x ) dx ≤ (2 π ) n . (1.10) If µ is discrete, there is equality if and only if f is a Gaussian. Otherwise, equalityholds if and only if f is proportional to a translation of the standard Gaussian. Note that the additional normalization of µ in Theorem 4 is critical due to thenon-homogeneity of the integral as well as the functional polarity with respect toAsplund scalar multiplication. Examples: (a) In 2006, Colesanti [ ] introduced the (Asplund) difference function ∆ ⋆ f of alog-concave function f ∈ LC c ( R n ) by ∆ ⋆ f = · f ⋆ · f , where f ( x ) = f ( − x ).By taking µ to be the even discrete probability measure δ ¯ e + δ − ¯ e ,concentrated on the stabilizer ¯ e ∈ S n − of SO( n −
1) and its antipodal,Theorem 4 reduces to a functional analogue of (1.3), Z R n f ( x ) dx Z R n (∆ ⋆ f ) ◦ ( x ) dx ≤ (2 π ) n (1.11)which can also be deduced from Ball’s functional Blaschke–Santal´o inequality(1.8) and the Pr´ekopa–Leindler inequality. Clearly, for even f , (1.11) coincideswith (1.8) which, thus, is a member of the family of inequalities of Theorem 4.(b) If µ is taken to be suitably normalized spherical Lebesgue measure σ , theinduced Asplund endomorphism Ψ σ has the following interesting properties: • Ψ σ f is radially symmetric for every f ∈ LC c ( R n ); • Ψ σ K = w ( K )2 · B n for every K ∈ K n .Moreover, we will see that inequality (1.10) for Ψ σ is strictly stronger thanthe functional analogue of Urysohn’s inequality (1.9) and, when restricted toindicators of convex bodies, yields a version of the Urysohn inequality (1.2)which is asymptotically sharp (see Section 4).The proof of Theorem 4 does not make use of Theorem 1, however, it followssimilar arguments as in the geometric setting, replacing the application of theBlaschke–Santal´o inequality (1.1) by its functional form. In particular, in Section 4we also prove a functional version of Theorem 2 showing that each of the inequalitiesof Theorem 4 is stronger than the one for the Asplund endomorphism Ψ σ and,hence, strictly stronger than the functional analogue of Urysohn’s inequality (1.9).For even µ , inequality (1.11) is proven to be the strongest one among the familyof inequalities (1.10). Finally, we will see that Theorem 1 can be recovered in anasymptotically optimal form by restricting Theorem 4 to indicators of convex bodies.6 . Background material In this section we recall additional basic facts from convex geometry and somenotions and results about convex and log-concave functions as well as the definitionof the convolution of spherical functions and measures. As general references, werecommend the monographs by Gardner [ ], Schneider [ ], Rockafellar [ ], andRockafellar and Wets [ ], as well as the survey [ ] by Colesanti.First recall that K n denotes the space of convex bodies in R n endowed withthe Hausdorff metric. Each K ∈ K n is uniquely determined by its support function h ( K, x ) = max { x · y : y ∈ K } , x ∈ R n , which is positively homogeneous of degree oneand subadditive. Conversely, every function on R n satisfying these two properties isthe support function of a unique convex body. For K, L ∈ K n , the support functionof their Minkowski sum K + L = { x + y : x ∈ K, y ∈ L } is given by h ( K + L, · ) = h ( K, · ) + h ( L, · ) (2.1)Moreover, for every ϑ ∈ SO( n ) and y ∈ R n , we have h ( ϑK, x ) = h ( K, ϑ − x ) and h ( K + y, x ) = h ( K, x ) + x · y (2.2)for all x ∈ R n . Also the Hausdorff distance d ( K, L ) of two convex bodies
K, L ∈ K n can be expressed conveniently by d ( K, L ) = k h ( K, · ) − h ( L, · ) k ∞ , where k · k ∞ denotes the maximum norm on C ( S n − ). Finally, recall that K ⊆ L if and only if h ( K, · ) ≤ h ( L, · ), in particular, h ( K, · ) > o ∈ int K .Recall that if K ∈ K n contains the origin in its interior, then K ◦ denotes thepolar body of K with respect to the origin. We will make frequent use of thefollowing polar coordinate formula for the volume of K ◦ , | K ◦ | = 1 n Z S n − h ( K, u ) − n du, (2.3)where integration is with respect to spherical Lebesgue measure. The mean width of a convex body K ∈ K n is defined by w ( K ) = 2 n | B n | Z S n − h ( K, u ) du. (2.4)The Steiner point s ( K ) ∈ R n of K is the unique point in relint K defined by s ( K ) = 1 | B n | Z S n − h ( K, u ) u du. (2.5)Recall that the mean width and the Steiner point are uniquely determinedby their Minkowski additivity and compatibility with rigid motions. To be moreprecise, a continuous map ̺ : K n → R is Minkowski additive and rigid motioninvariant if and only if it is a constant multiple of the mean width w , while theSteiner point is the unique continuous map s : K n → R n which is Minkowskiadditive and rigid motion equivariant (cf. [ , Section 3.3]).7ssociated with each convex body K ∈ K n is a finite non-negative Borel measure S ( K, · ) on S n − , its area measure of order one . Using the Laplacian (or Laplace–Beltrami operator) ∆ S on S n − , it can be defined by the relation (to be understoodin a distributional sense) S ( K, · ) = 1 n − S h ( K, · ) + h ( K, · ) . As shown by Weil [ ], the set { S ( K, · ) : K ∈ K n } is not dense in the set of allnon-negative measures on S n − with barycenter at the origin in the weak topology.Each non-negative even measure µ on S n − generates a uniquely determinedorigin-symmetric convex body Z µ ∈ K n by h ( Z µ , u ) = Z S n − | u · v | dµ ( v ) , u ∈ S n − . (2.6)The bodies obtained in this way constitute the class of origin-symmetric zonoids ,which naturally arise also in various other contexts (see, e.g., [ , Chapter 3.5]).We turn now to convex and log-concave functions on R n . Let Cvx( R n ) denotethe set of convex and lower semi-continuous functions ϕ : R n → ( −∞ , ∞ ] whichare proper , that is, not identically + ∞ . Two convex sets naturally associated toany ϕ ∈ Cvx( R n ) are its domain , dom ϕ = { x ∈ R n : ϕ ( x ) < + ∞} , and its epigraph defined by epi ϕ = { ( x, ξ ) ∈ R n × R : ϕ ( x ) ≤ ξ } . Note that for any ϕ ∈ Cvx( R n ), dom ϕ is non-empty and epi ϕ is closed and non-empty. We call afunction ϕ ∈ Cvx( R n ) coercive if lim k x k→∞ ϕ ( x ) = + ∞ and we denote by Cvx c ( R n )the set of all coercive ϕ ∈ Cvx( R n ). We also note that (see, e.g., [ , Lemma 2.5]), ϕ ∈ Cvx( R n ) is coercive if and only if there exist γ > β ∈ R such that forevery x ∈ R n , ϕ ( x ) ≥ γ k x k + β. (2.7)Next we endow the spaces Cvx( R n ) and Cvx c ( R n ) with the topology induced byepi-convergence. Recall that a sequence of ϕ k ∈ Cvx( R n ) is called epi-convergent to ϕ : R n → ( −∞ , ∞ ] if for all x ∈ R n the following two conditions hold: • ϕ ( x ) ≤ lim inf k →∞ ϕ k ( x k ) for every sequence x k that converges to x . • There exists a sequence x k converging to x such that ϕ ( x ) = lim k →∞ ϕ k ( x k ).In this case, we write ϕ k epi → ϕ . Note that the limiting function ϕ is again convexand lower semi-continuous. However, in general, ϕ need not be proper. Lemma 2.1 ([ , Theorem 7.17]) If ϕ, ϕ k ∈ Cvx( R n ) and int dom ϕ is non-empty,then the following statements are equivalent to ϕ k being epi-convergent to ϕ :(i) There exists a dense set D ⊆ R n such that ϕ k ( x ) → ϕ ( x ) for every x ∈ D .(ii) The sequence ϕ k converges uniformly to ϕ on every compact subset of R n thatdoes not intersect the boundary of dom ϕ . , Section 7.B]).For ϕ, ψ ∈ Cvx( R n ), their infimal convolution is defined by( ϕ (cid:3) ψ )( x ) = inf x + x = x { ϕ ( x ) + ψ ( x ) } . If ϕ (cid:3) ψ does not attain the value −∞ , then it is convex, proper, andepi( ϕ (cid:3) ψ ) = epi ϕ + epi ψ. However, ϕ (cid:3) ψ need not be semi-continuous. A quite useful condition to ensurelower semi-continuity of the infimal convolution of ϕ, ψ ∈ Cvx( R n ) can be foundin [ , Corollary 9.2.2] and requires thatlim λ →∞ ϕ ( y + λx ) λ + lim λ →∞ ψ ( z − λx ) λ > x ∈ R n and arbitrary y ∈ dom ϕ , z ∈ dom ψ .For t > ϕ ∈ Cvx( R n ), the Moreau envelope e t ϕ of ϕ is defined by e t ϕ = ϕ (cid:3) t k · k . For the proof of Theorem 3, we require the following of its simple properties.
Lemma 2.2 ( [ , Theorems 1.25 & 2.26]) Suppose that ϕ ∈ Cvx( R n ) . Then thefollowing statements hold:(i) e t ϕ ∈ Cvx( R n ) and it is finite for every t > ;(ii) e t ϕ ( x ) converges to ϕ ( x ) monotonously from below for every x ∈ R n as t ց .In particular, e t ϕ epi → ϕ as t ց . Now, let LC( R n ) = { f = e − ϕ : ϕ ∈ Cvx( R n ) } denote the set of all proper,log-concave, and upper semi-continuous functions on R n and recall thatLC c ( R n ) = (cid:26) f ∈ LC( R n ) : lim k x k→∞ f ( x ) = 0 (cid:27) = (cid:8) f = e − ϕ : ϕ ∈ Cvx c ( R n ) (cid:9) . We call a sequence f k = e − ϕ k ∈ LC( R n ) or LC c ( R n ), respectively, epi-convergent to f = e − ϕ , if ϕ k ∈ Cvx( R n ) or Cvx c ( R n ), respectively, epi-converges to ϕ .Next, recall that the Asplund sum f ⋆ g of f = e − ϕ , g = e − ψ ∈ LC( R n ) is relatedto the infimal convolution ϕ (cid:3) ψ of ϕ, ψ ∈ Cvx( R n ) by f ⋆ g = e − ϕ (cid:3) ψ . (2.9)In particular, since Cvx( R n ) is not closed under infimal convolution, the spaceLC( R n ) is not closed under Asplund addition. However, as the following lemmashows, this is no longer the case when considering coercive functions.9 emma 2.3 Suppose that f, g ∈ LC c ( R n ) , a, b > and y ∈ R n . Then the followingstatements hold:(i) f ⋆ g ∈ LC c ( R n ) ;(ii) ( a · f ) ⋆ ( b · f ) = ( a + b ) · f ;(iii) ( f ⋆ { y } )( x ) = f ( x − y ) .Proof. In order to prove (i), let f = e − ϕ and g = e − ψ with ϕ, ψ ∈ Cvx c ( R n ). Then,by (2.9), it is sufficient to show that ϕ (cid:3) ψ ∈ Cvx c ( R n ). First note that since ϕ and ψ are coercive, they are bounded from below and, consequently, so is ϕ (cid:3) ψ which is,therefore, convex and proper. Moreover, from the definition of infimal convolutionand the triangle inequality, it follows that ϕ (cid:3) ψ is coercive. It remains to show that ϕ (cid:3) ψ is lower semi-continuous, for which we check that condition (2.8) is satisfied.To this end, we use (2.7) to conclude that there exist γ ϕ , γ ψ > β ϕ , β ψ ∈ R such that ϕ ( w ) ≥ γ ϕ k w k + β ϕ and ψ ( w ) ≥ γ ψ k w k + β ψ for every w ∈ R n . Thus,lim λ →∞ ϕ ( y + λx ) λ + lim λ →∞ ψ ( z − λx ) λ ≥ γ ϕ k x k + γ ψ k x k > x ∈ R n which completes the proof of (i). Statements (ii) and(iii) follow easily from the definition of the Asplund sum and multiplication. (cid:4) Examples: (a) If K ∈ K n , then the indicator function K ∈ LC c ( R n ) is defined by K ( x ) = (cid:26) x ∈ K, K ∈ K n containing the origin in its interior, its gauge or Minkowski functional is given by k x k K = min { λ ≥ x ∈ λK } , x ∈ R n . If K is origin-symmetric, k · k K is the norm with unit ball K . For K = B n , wesimply write k · k instead of k · k B n . Another interesting class of log-concavefunctions consists of those f = e − ϕ ∈ LC c ( R n ), where ϕ = 1 p k · k pK , p ≥ . (2.10)In particular, f ∈ LC c ( R n ) is called a Gaussian if there exist a > y ∈ R n and an origin-symmetric ellipsoid E ⊆ R n such that f ( x ) = a e − k x − y k E , x ∈ R n . For a = (2 π ) − n/ , y = o and E = B n , we obtain the standard Gaussian ψ n .10et us mention here another useful volume formula for convex bodies, involvingthe functions defined in (2.10). If K ∈ K n contains the origin in its interior, then | K | = 1 p n/p Γ (cid:16) np (cid:17) Z R n exp (cid:18) − p k x k pK (cid:19) dx. (2.11)The Legendre transform L : Cvx( R n ) → Cvx( R n ) is defined by( L ϕ )( x ) = sup y ∈ R n x · y − ϕ ( y ) , x ∈ R n . It is a classical notion with many applications in several areas which are extensivelycovered in the literature (e.g., [ , ]). We collect a number of its well knownproperties for quick later reference in the following proposition. Note that theproperties from (i) were recently shown to essentially characterize the Legendretransform in a fundamental paper by Artstein-Avidan and Milman [ ]. Proposition 2.4 (see, e.g., [ , ]) For ϕ k , ϕ, ψ ∈ Cvx( R n ) and K ∈ K n , thefollowing statements hold:(i) LL ϕ = ϕ and if ϕ ≤ ψ , then L ϕ ≥ L ψ ;(ii) ϕ is coercive if and only if dom L ϕ contains the origin in its interior;(iii) ϕ k epi → ϕ if and only if L ϕ k epi → L ϕ ;(iv) L ( − log K ) = h ( K, · ) and if < p, q < ∞ are such that p + q = 1 and K contains the origin in its interior, then L (cid:18) p k · k pK (cid:19) = 1 q k · k qK ◦ . The Legendre transform gives rise to several constructions for log-concavefunctions. Recall that for f ∈ LC( R n ), its support function is h ( f, · ) = L ( − log f )and note that h ( f, · ) ∈ Cvx( R n ). For our purposes it is particularly useful to observethat support functions are Asplund additive (cf. [ , p.518]), in the sense that for f, g ∈ LC c ( R n ), we have h ( f ⋆ g, · ) = h ( f, · ) + h ( g, · ) (2.12)which is an extension of (2.1) to LC c ( R n ). Moreover, for ϑ ∈ SO( n ) and y ∈ R n , h ( ϑf, · ) = h ( f, ϑ − x ) and h ( f ( . − y ) , x ) = h ( f, x ) + x · y (2.13)for every x ∈ R n – an extension of properties (2.2) to LC c ( R n ).Recall that for f ∈ LC( R n ), its polar function is defined by f ◦ = e −L ( − log f ) .From the definition of the Legendre transform and Proposition 2.4, one obtains:11 emma 2.5 For f, g ∈ LC c ( R n ) , a > , and K ∈ K n containing the origin in itsinterior, the following statements hold:(i) ( f ◦ ) ◦ = f ;(ii) ( Af ) ◦ = A − T f ◦ for every A ∈ GL( n ) ;(iii) ( f ⋆ g ) ◦ = f ◦ g ◦ and ( a · f ) ◦ = ( f ◦ ) a ;(iv) ( K ) ◦ = e −k·k K ◦ and if < p, q < ∞ are such that p + q = 1 , then (cid:16) e − p k·k pK (cid:17) ◦ = e − q k·k qK ◦ . We next state a more general version of the functional Blaschke–Santal´oinequality due to Artstein-Avidan, Klartag, and Milman [ ] required in the proofof Theorem 4. We restrict ourselves to log-concave functions and recall that thecentroid of an integrable function f on R n such that R R n f dx > f = R R n xf ( x ) dx R R n f ( x ) dx . Theorem 2.6 (Artstein-Avidan et al. [ ]) Suppose that f ∈ LC( R n ) is such that < R R n f ( x ) dx < ∞ and let ˜ f ( x ) = f ( x − cent f ) . Then Z R n f ( x ) dx Z R n ˜ f ◦ ( x ) dx ≤ (2 π ) n with equality if and only if f is a Gaussian. Let us note that for K ∈ K n containing the origin in its interior and f = e − k·k K in Theorem 2.6, one recovers a version of the geometric Blaschke–Santal´o inequalityequivalent to (1.1).In the final part of this section, we recall the convolution of spherical functionsand measures and its extension to functions on R n . We are particularly interestedin convolutions with zonal measures, that is, SO( n −
1) invariant measures on S n − ,where SO( n −
1) is the subgroup of SO( n ) keeping an arbitrary pole ¯ e ∈ S n − fixed.First, recall that since SO( n ) is a compact Lie group, the convolution τ ∗ µ of signedmeasures τ, µ on SO( n ) can be defined by Z SO( n ) f ( ϑ ) d ( τ ∗ µ )( ϑ ) = Z SO( n ) Z SO( n ) f ( ηθ ) dτ ( η ) dµ ( θ ) , f ∈ C (SO( n )) . Since S n − is diffeomorphic to the homogeneous space SO( n ) / SO( n − S n − andright SO( n −
1) invariant functions and measures on SO( n ), respectively. Usingthis correspondence, the convolution of measures on SO( n ) induces an associativeconvolution product of measures on S n − (cf. [ ] for more details).12ere, we only note that the spherical convolution with a zonal measure on S n − takes an especially simple form. If for u ∈ S n − , we denote by ϑ u ∈ SO( n ) anarbitrary rotation such that ϑ u ¯ e = u , the convolution of h ∈ C ( S n − ) and a zonalmeasure µ on S n − is given by( h ∗ µ )( u ) = Z S n − h ( ϑ u v ) dµ ( v ) , u ∈ S n − . (2.14)Since ϑ ηu = ηϑ u for any η ∈ SO( n ) up to right-multiplication by an element ofSO( n − ηh ) ∗ µ = η ( h ∗ µ ). It is also not difficultto check that the convolution (2.14) is selfadjoint and that spherical convolution ofzonal functions and measures is Abelian.Motivated by (2.14) and the significance of the spherical convolution ∗ forMinkowski endomorphisms, we now introduce an extension of ∗ to the followingimportant open subset of convex functions in Cvx( R n ),Cvx ( o ) ( R n ) = { ϕ ∈ Cvx( R n ) : o ∈ int dom ϕ } . Note that, by Proposition 2.4 (ii), ϕ ∈ Cvx ( o ) if and only if L ϕ ∈ Cvx c ( R n ) or,equivalently, f ∈ LC c ( R n ) if and only if h ( f, · ) ∈ Cvx ( o ) ( R n ) . (2.15)In the following, for x ∈ R n \{ o } , let ϑ x ∈ SO( n ) denote an arbitrary rotation suchthat ϑ x ¯ e = x k x k . Definition.
Suppose that ϕ ∈ Cvx ( o ) ( R n ) and let µ be a non-negative zonalmeasure on S n − . The convolution ϕ ⊛ µ is defined for x ∈ R n \{ o } by( ϕ ⊛ µ )( x ) = Z S n − ϕ ( k x k ϑ x v ) dµ ( v ) (2.16)and at the origin by ( ϕ ⊛ µ )( o ) = lim inf k x k→ ( ϕ ⊛ µ )( x ) .We will show in the next section that ϕ ⊛ µ is indeed a well defined function inCvx ( o ) ( R n ), but already note here that if ϕ is homogeneous of some degree p ∈ R ,then, by (2.14), ϕ ⊛ µ coincides with the homogeneous extension of degree p of b ϕ ∗ µ to R n , where b ϕ is the restriction of ϕ to S n − .
3. Minkowski and Asplund endomorphisms
In the following we first review additional background material on Minkowskiendomorphisms required in the proofs of Theorems 1 and 2 and the discussionthereof in Section 4. In the second part of this section we give the proof of Theorem 3as well as that of an auxiliary result used in the proof of an analogue of Theorem 2for log-concave functions. 13e begin by recalling a notion of monotonicity for operators on convex bodieswhich is of particular importance for Minkowski endomorphisms.
Definition
A map Φ : K n → K n is called weakly monotone if Φ K ⊆ Φ L for all K, L ∈ K n such that K ⊆ L and s ( K ) = s ( L ) = o .The quest to establish a classification of all Minkowski endomorphisms has itsorigin in the paper [ ] from 1974 by Schneider. The following result – combiningtheorems by Dorrek and Kiderlen – represents the status quo on this difficult taskwhich has not yet been completed. Here, we call a measure on S n − linear if it hasa density (w.r.t. spherical Lebesgue measure) of the form u x · u for some x ∈ R n . Theorem 3.1 (Dorrek [ ], Kiderlen [ ]) If Φ : K n → K n is a Minkowskiendomorphism, then there exists a signed zonal measure µ on S n − with centerof mass at the origin such that h (Φ K, · ) = h ( K, · ) ∗ µ (3.1) for every K ∈ K n . The measure µ is uniquely determined by Φ .Moreover, Φ is monotone if and only if µ is non-negative and Φ is weakly monotoneif and only if µ is non-negative up to addition of a linear measure. The measure µ uniquely associated with the Minkowski endomorphism Φ viathe relation (3.1) is called the generating measure of Φ and we frequently indicate itby writing Φ µ . Exploiting this one-to-one correspondence, we can endow the cone ofMinkowski endomorphisms with the topology induced by the weak convergence oftheir generating measures. Before we exhibit some prominent examples, let us notethat for n = 2, Theorem 3.1 is due to Schneider [ ] who also showed that in thisspecial case all Minkowski endomorphisms are weakly monotone. The conjecturethat the same is true for n ≥ ]. Examples: (a) Recall that σ denotes the SO( n ) invariant probability measure on S n − . TheMinkowski endomorphism Φ σ : K n → K n generated by σ satisfiesΦ σ K = w ( K )2 B n (3.2)for every K ∈ K n and, thus, the inequality | K || Φ ◦ σ | ≤ | B n | is precisely theUrysohn inequality.(b) The unique discrete zonal probability measure on S n − with center of mass atthe origin is given by ν = 12 ( δ ¯ e + δ − ¯ e ) . (3.3)It is the generating measure of the central symmetrization ∆ : K n → K n .14c) The generating measure of the Minkowski endomorphism Π : K n → K n (recall our normalization Π B n = B n ) is the invariant probability measure σ ¯ e ⊥ concentrated on the equator S n − ∩ ¯ e ⊥ . Noting that σ ¯ e ⊥ = S (cid:0) [ − ¯ e, ¯ e ] , · (cid:1) , Berg and the second author [ ] considered, more generally, Minkowskiendomorphisms generated by measures of the form S ( Z, · ) for some zonoid Z ∈ K n and established Theorem 1 for such maps. However, this class is notdense in all monotone Minkowski endomorphisms.(d) The Minkowski endomorphism J : K n → K n , defined byJ K = K − s ( K ) , (3.4)is weakly monotone and its generating measure is given by δ ¯ e − n (¯ e · . ) dσ .For the reader’s convenience, we prove next two useful previously knownproperties of Minkowski endomorphisms required in the proof of Theorem 2. Lemma 3.2
Suppose that
Φ : K n → K n is a Minkowski endomorphism withgenerating measure µ on S n − . Then the following statements hold:(i) w (Φ K ) = µ ( S n − ) w ( K ) for every K ∈ K n ;(ii) If Φ is non-trivial and weakly monotone and K ∈ K n has non-empty interior,then Φ K contains the origin in its interior.Proof. In order to see (i), we use that the spherical convolution is selfadjoint andAbelian for zonal measures. These facts combined with (2.4) and (3.2) yield, w (Φ K ) = 2 Z S n − ( h ( K, · ) ∗ µ )( u ) dσ ( u ) = 2 Z S n − ( h ( K, · ) ∗ σ )( u ) dµ ( u )= w ( K ) Z S n − h ( B n , u ) dµ ( u ) = µ ( S n − ) w ( K )for every K ∈ K n . For the proof of (ii), first note that every non-trivial Minkowskiendomorphism maps Euclidean balls of positive radii to origin-symmetric balls ofpositive radii by (i). Now, using that the Steiner point s ( K ) ∈ int K for every K ∈ K n with non-empty interior, we obtain (ii) from the translation invariance ofΦ, which implies Φ K = Φ( K − s ( K )), and its monotonicity on bodies with Steinerpoints at the origin. (cid:4) We turn now to Asplund endomorphisms and prove the following critical resultat the core of Theorem 3. 15 roposition 3.3
Suppose that ϕ ∈ Cvx ( o ) ( R n ) . Then ϕ ⊛ µ is a well definedfunction in Cvx ( o ) ( R n ) . Moreover, the map ϕ ϕ ⊛ µ defines a continuous linearoperator from Cvx ( o ) ( R n ) to itself which commutes with the action of SO( n ) .Proof. First note that the right hand side of (2.16) is independent of the choice of ϑ x by the SO( n −
1) invariance of µ . Since ϕ is convex on R n , it is bounded frombelow by an affine function and, thus, the negative part of ϕ is bounded on everysphere in R n . Consequently, the integral in (2.16) is well defined and takes valuesin ( −∞ , ∞ ]. Moreover, since o ∈ int dom ϕ , there exists r > ϕ takesfinite values on rB n , which implies that also ϕ ⊛ µ takes finite values on rB n . Inparticular, ϕ ⊛ µ is proper and o ∈ int dom ϕ ⊛ µ .The proof that ϕ ⊛ µ is convex on R n is rather tedious and technical and wetherefore postpone it to the Appendix. In order to see that ϕ ⊛ µ ∈ Cvx ( o ) ( R n ), itremains to show that it is lower semi-continuous. To this end, let x ∈ R n \{ o } andnote that the values of ϕ on any sphere are larger than the minimum of an affinefunction below ϕ on the same sphere. Hence, we may use Fatou’s Lemma and thesemi-continuity of ϕ to conclude thatlim inf x → x ( ϕ ⊛ µ )( x ) ≥ Z S n − lim inf x → x ϕ ( k x k ϑ x u ) dµ ( u ) ≥ Z S n − ϕ ( k x k ϑ x u ) dµ ( u ) = ( ϕ ⊛ µ )( x )which combined with our definition of ( ϕ ⊛ µ )( o ) yields the lower semi-continuityof ϕ ⊛ µ on R n and completes the proof that ϕ ⊛ µ ∈ Cvx ( o ) ( R n ).Since the linearity and commutativity with respect to the action of SO( n ) of themap ϕ ϕ ⊛ µ on Cvx ( o ) ( R n ) are immediate consequences of the definition of ⊛ ,it only remains to show that this map is continuous with respect to the topologyinduced by epi-convergence. Therefore, let ϕ k ∈ Cvx ( o ) ( R n ) be an epi-convergentsequence with limit ϕ ∈ Cvx ( o ) ( R n ). In order to prove that ϕ k ⊛ µ epi → ϕ ⊛ µ (3.5)we proceed in three steps. First we claim that the Moreau envelope e t ϕ of ϕ satisfies e t ϕ ⊛ µ epi → ϕ ⊛ µ (3.6)as t ց
0. Indeed, by (2.16), Lemma 2.2 (ii) and the monotone convergence theorem,lim t ց ( e t ϕ ⊛ µ )( x ) = lim t ց Z S n − e t ϕ ( k x k ϑ x v ) dµ ( v ) = Z S n − ϕ ( k x k ϑ x v ) dµ ( v ) = ( ϕ ⊛ µ )( x )for every x ∈ R n , which, by Lemma 2.1 (i), implies (3.6). In a second step, letting k → ∞ , we claim that for every t > e t ϕ k ⊛ µ epi → e t ϕ ⊛ µ. (3.7)16o see this, we first note that e t ϕ k is epi-convergent to e t ϕ as k → ∞ , byProposition 2.4 (iii) and the fact that the Legendre transform maps infimalconvolution to pointwise addition. Moreover, by Lemma 2.2 (i), e t ϕ k and e t ϕ areboth finite and, hence, their epi-convergence is equivalent to uniform convergenceon compact subsets of R n , by Lemma 2.1 (ii), which in turn is preserved under theconvolution ⊛ with the measure µ .Finally, we show that, letting k → ∞ ,( ϕ k ⊛ µ )( x ) → ( ϕ ⊛ µ )( x ) for every (cid:26) x ∈ int dom ( ϕ ⊛ µ ) \{ o } ,x / ∈ cl dom ( ϕ ⊛ µ ) , (3.8)which, by Lemma 2.1 (i) and the fact that R n without the boundary of dom ( ϕ ⊛ µ )and the origin is a dense subset, concludes the proof of (3.5) and the proposition.To this end, first suppose that x cl dom ( ϕ ⊛ µ ). Then there exists a closedball B such that x ∈ B and B ∩ cl dom ( ϕ ⊛ µ ) = ∅ . By (3.6) and Lemma 2.1 (ii), e t ϕ ⊛ µ converges to ϕ ⊛ µ uniformly on B as t ց
0. Thus, for every c >
0, we canfind t > t ≤ t , we have e t ϕ ⊛ µ ≥ c on B . Similarly, by (3.7)and Lemma 2.1 (ii), we can find k ∈ N such that for all k ≥ k , e t ϕ k ⊛ µ > c on B .Since e t ϕ k ≤ ϕ k , by Lemma 2.2 (ii), and the convolution ⊛ is obviously monotone, ϕ k ⊛ µ > c on B for all k ≥ k . Since c > ϕ k ⊛ µ converges to infinity uniformly on B , which proves (3.8) for x cl dom ( ϕ ⊛ µ ).Suppose now that x ∈ int dom ( ϕ ⊛ µ ) is non-zero and fix a rotation ϑ x ∈ SO( n )such that ϑ x ¯ e = x k x k . Then there exists an open ε -ball B ε ⊆ int dom ( ϕ ⊛ µ ) centeredat x . By (2.16), ( ϕ k ⊛ µ )( x ) and ( ϕ ⊛ µ )( x ) are determined by integrating the valuesof ϕ k and ϕ , respectively, over k x k S n − . Therefore, we consider the compact set C = ( k x k S n − ) ∩ ( R n \ int dom ϕ ) . If C is empty, then, by Lemma 2.1 (ii), ϕ k converges uniformly to ϕ on k x k S n − and, consequently, by (2.16), we conclude that ( ϕ k ⊛ µ )( x ) → ( ϕ ⊛ µ )( x ).Thus, assume that there exists some y ∈ C . Since o ∈ int dom ϕ , there existsan open δ -ball B δ ⊆ dom ϕ centered at o . Since dom ϕ is convex, each ray through y ∈ C emanating from a point in B δ intersects the boundary of dom ϕ in exactly onepoint. In particular, the parts of these rays starting at y are completely containedin R n \ dom ϕ . Hence, for every y ∈ C , there exists an open cone C y with apex y contained in int ( R n \ dom ϕ ) and intersecting R S n − , for any R > k x k , in an opencap whose diameter depends only on R and δ .Choosing R = k x k + ε , we have x = R k x k x ∈ B ε , that is, k x k = R and( ϕ ⊛ µ )( x ) < ∞ which imply, by (2.16), that µ ( { u ∈ S n − : ϕ ( Rϑ x u ) = ∞} ) = 0 . Consequently, since ϕ is infinite on C y ∩ R S n − , we have µ ( { u ∈ S n − : Rϑ x u ∈ C y } ) = 0 . Rϑ x u ∈ C y if and only if k x k ϑ x u ∈ k x k R ( C y ∩ R S n − ) ⊆ k x k S n − , we inferthat for every y ∈ C , there exists an open subset U y of S n − of µ -measure zero suchthat k x k ϑ x U y is an open neighborhood of y . The family ( k x k ϑ x U y ) y ∈ C is an opencover of the compact set C , hence, there exists a finite subcover ( k x k ϑ x U y i ) mi =1 and,by the sub-additivity of µ , we have µ ( U ) = 0 for U = S mi =1 U y i .Since the compact set C ′ = k x k S n − \k x k ϑ x U is disjoint from bd dom ϕ , ϕ k converges uniformly to ϕ on C ′ , by Lemma 2.1 (ii). For every b ε >
0, we thus find k such that for all k ≥ k we have | ϕ k ( z ) − ϕ ( z ) | ≤ b ε for all z ∈ C ′ . Consequently,( ϕ k ⊛ µ )( x ) = Z S n − ϕ k ( k x k ϑ x v ) dµ ( v ) = Z S n − \ U ϕ k ( k x k ϑ x v ) dµ ( v ) ≤ Z S n − \ U ( ϕ ( k x k ϑ x v ) + b ε ) dµ ( u ) = ( ϕ ⊛ µ )( x ) + b ε µ ( S n − ) , that is, ( ϕ k ⊛ µ )( x ) is finite for all k ≥ k . Hence, we can infer that | ( ϕ k ⊛ µ )( x ) − ( ϕ ⊛ µ )( x ) | ≤ Z S n − \ U | ϕ k ( k x k ϑ x v ) − ϕ ( k x k ϑ x v ) | dµ ( v ) ≤ b ε µ ( S n − ) , which completes the proof of (3.8). (cid:4) We are now in a position to complete the proof of Theorem 3.
Proof of Theorem 3.
It follows from (2.15) and Proposition 3.3 that Ψ µ f is welldefined for every f ∈ LC c ( R n ). By (2.12) and (2.16), Ψ µ : LC c ( R n ) → LC c ( R n )is Asplund additive. By Lemma 2.3 (iii), f ⋆ { y } coincides with the translate of f ∈ LC c ( R n ) by y ∈ R n . Hence, using Ψ µ { y } = Φ µ { y } = { o } , where Φ µ denotesthe Minkowski endomorphism generated by µ , we deduce thatΨ µ ( f ⋆ { y } ) = Ψ µ f ⋆ Φ µ { y } = Ψ µ f ⋆ { o } = Ψ µ f. The commutativity of Ψ µ with the action of SO( n ) follows from the definitions of ⊛ and the support function h ( f, · ) and the fact that the Legendre transform commuteswith the action of SO( n ) on Cvx( R n ). Continuity of Ψ µ : LC c ( R n ) → LC c ( R n ) isa consequence of Lemma 2.4 (iii) and Proposition 3.3. The monotonicity of Ψ µ follows from the monotonicity of support functions and that of ⊛ .By Proposition 2.4 (iv) and the remark after definition (2.16), Ψ µ K = Φ µ K for every K ∈ K n . Thus, µ is uniquely determined by Ψ µ by Theorem 3.1. (cid:4) The classification of additive (in a set-theoretic sense) maps on convex andlog-concave functions has recently become the focus of intensive investigations(see, e.g., [ , , , , ]). It is certainly an interesting open problemwhether there exist additional monotone Asplund endomorphisms different fromthe ones provided by Theorem 3 and, if so, whether the property that indicators ofconvex bodies are mapped to indicators, satisfied by the Asplund endomorphismsΨ µ from Theorem 3, may help to characterize them.18e conclude this section with a functional analogue of Lemma 3.2 (i). Lemma 3.4 If µ is a non-negative zonal measure on S n − with center of mass atthe origin, then Z R n h (Ψ µ f, x ) dγ n ( x ) = µ ( S n − ) Z R n h ( f, x ) dγ n ( x ) (3.9) for every f ∈ LC c ( R n ) .Proof. Using polar coordinates and the density ψ n of the Gaussian measure, yields Z R n h (Ψ µ f, x ) dγ n ( x ) = n | B n | Z ∞ Z S n − ( h ( f, · ) ⊛ µ )( ru ) ψ n ( ru ) r n − dσ ( u ) dr. By the SO( n ) invariance of ψ n , relation (3.9) follows, if we can show that Z S n − ( h ⊛ µ )( ru ) dσ ( u ) = µ ( S n − ) Z S n − h ( ru ) dσ ( u ) (3.10)for every h ∈ Cvx ( o ) ( R n ) and every r >
0. To this end, first assume that dom h = R n and, hence, that h is continuous. Since ( h ⊛ µ )( ru ) = ( h ( r · ) ∗ µ )( u ) for every u ∈ S n − , we obtain, as in the proof of Lemma 3.2 (i), from the fact that sphericalconvolution is selfadjoint and Abelian for zonal measures, that Z S n − ( h ⊛ µ )( ru ) dσ ( u ) = Z S n − ( h ( r · ) ∗ σ )( u ) dµ ( u ) . By the SO( n ) invariance of σ , ( h ( r · ) ∗ σ )( u ) is independent of u and (3.10) follows.For general h ∈ Cvx ( o ) ( R n ), we use that the Moreau envelope e t h of h is convexand finite and converges monotonously to h , by Lemma 2.2. By what we haveshown above, (3.10) holds for e t h for every t >
0. Hence, by monotone convergence,we conclude that (3.10) holds generally. (cid:4)
4. Proof of the main results
In this section we first prove Theorem 2 and deduce Theorem 1 from it. Wethen show that a further extension of Theorem 1 to all merely weakly monotoneMinkowski endomorphisms is not possible. In order to prove Theorem 4, we willestablish first a counterpart of Theorem 2 for log-concave functions. We concludethis section by showing that each of the inequalities from Theorem 4 is strictlystronger than the functional analogue of Urysohn’s inequality.
Proof of Theorem 2.
Let K ∈ K n have non-empty interior and note that thenormalization Φ B n = B n ensures that there is equality in both inequalities of(1.6) if K is a Euclidean ball. By Theorem 3.1, the generating measure µ of Φ isnon-negative and, by our normalization, 1 = h (Φ B n , · ) = µ ( S n − ). Moreover, byLemma 3.2 (ii), Φ K contains the origin in its interior.19n order to establish the left hand inequality of (1.6), we use the polar coordinateformula for volume (2.3), Jensen’s inequality, and (2.4) to obtain (cid:18) | Φ ◦ K || B n | (cid:19) − /n = (cid:18)Z S n − h (Φ K, u ) − n dσ ( u ) (cid:19) − /n ≤ Z S n − h (Φ K, u ) dσ ( u ) = w (Φ K )2 . An application of Lemma 3.2 (i) now yields the desired inequality. By the equalityconditions for Jensen’s inequality, equality holds here, and thus in the left hand sideof (1.6), if and only if h (Φ K, · ) is constant, that is, if and only if Φ K is a ball.For the proof of the right hand inequality of (1.6), we may assume, by thetranslation invariance of both sides, that s ( K ) = o , which implies h ( K, · ) > K s = K ◦ . First, we use the polar coordinate formula for volume (2.3), (2.14), andJensen’s inequality (note that µ is a probability measure) to obtain | Φ ◦ K | = 1 n Z S n − h (Φ K, u ) − n du = 1 n Z S n − (cid:18)Z S n − h ( K, ϑ u v ) dµ ( v ) (cid:19) − n du ≤ n Z S n − Z S n − h ( K, ϑ u v ) − n dµ ( v ) du. Since Φ and the polar map commute with SO( n ) transforms, we may replace K hereby a rotated copy θK and integrate over SO( n ) with respect to the Haar measure,to arrive at | Φ ◦ K | = Z SO( n ) | Φ ◦ ( θK ) | dθ ≤ n Z SO( n ) Z S n − Z S n − h ( K, θ − ϑ u v ) − n dµ ( v ) du dθ, (4.1)where we also used (2.2) in the last step. By using Fubini’s theorem twice, theinvariance of the Haar measure on SO( n ), and the fact that µ ( S n − ) = 1, we obtainthe desired inequality, | Φ ◦ K | ≤ n Z S n − Z SO( n ) h ( K, θ − u ) − n dθ du = Z SO( n ) | θK ◦ | dθ = | K ◦ | . By the above arguments, equality holds in the right hand inequality of (1.6) ifand only if we have equality in (4.1). By the equality condition of Jensen’s inequalitythis is the case if and only if for almost every u ∈ S n − and almost every θ ∈ SO( n )there exist c u,θ ∈ R + such that h ( K, θ − ϑ u v ) = c u,θ for µ -a.e. v ∈ S n − . Clearly,this is the case if and only if for every η ∈ SO( n ) (by the continuity of h ( K, · )),there exist c η ∈ R + such that h ( K, ηv ) = c η for µ -a.e. v ∈ S n − . (4.2)Let A η = { v ∈ S n − : h ( K, ηv ) = c η } ⊆ S n − and note that µ ( S n − \ A η ) = 0. Notethat A η is closed, by the continuity of h ( K, · ), and contains the support of µ . If thelatter was not the case, we could find a v ∈ supp µ \ A η and an open neighborhoodof v which does not intersect A η . As v ∈ supp µ this neighborhood has positive µ -measure, a contradiction. 20f µ is discrete it must coincide with the measure ν given by (3.3) and, hence,Φ = ∆. Since supp ν = {− ¯ e, ¯ e } , it follows immediately from (4.2) that h ( K, · ) takesthe same value on antipodal points, that is, K is origin-symmetric.It remains to be shown that if µ is not discrete, then (4.2) holds if and only if h ( K, · ) is constant on S n − , or equivalently, if K is an origin-symmetric Euclideanball. Since µ is non-zero and not discrete, there exists w ∈ supp µ \{− ¯ e, ¯ e } . By theSO( n − µ , the entire parallel subsphere orthogonal to ¯ e through w iscontained in supp µ . Hence, by (4.2), h ( K, ηv ) = c η for every η ∈ SO( n ) and all v inthis subsphere. Choosing η ′ such that this subsphere and a copy of it rotated by η ′ intersect, we see that the value of h ( K, · ) at the intersection is given by c id and c η ′ ,thus, these values must be equal. By repeating this argument finitely many times,we can reach every point on S n − implying that h ( K, · ) is constant as desired. (cid:4) Theorem 1 is now an easy consequence of the right-hand inequality of (1.6).
Proof of Theorem 1.
Let K ∈ K n have non-empty interior and assume w.l.o.g. thatΦ B n = B n . Then, by the right-hand inequality of (1.6) and the Blaschke–Santal´oinequality (1.1), it follows that | K || Φ ◦ K | ≤ | K || K s | ≤ | B n | = | B n || Φ ◦ B n | . The equality | K || Φ ◦ K | = | B n | holds if and only if equality holds both in the right-hand inequality of (1.6) and the Blaschke–Santal´o inequality (1.1), that is, if andonly if Φ = ∆ and K is an ellipsoid or if K is a Euclidean ball. (cid:4) Next, we want to show that an extension of Theorem 1 to all merely weaklymonotone Minkowski endomorphisms is impossible.
Theorem 4.1
For every n ≥ , the volume product | K || J ◦ K | is unbounded for theweakly monotone Minkowski endomorphism J : K n → K n , J K = K − s ( K ) .Proof. We begin with dimension n = 2, where for c >
0, we consider the triangle K c ∈ K of unit volume defined by K c = conv (cid:8) ( c, , (cid:0) , c (cid:1) , (cid:0) , − c (cid:1)(cid:9) . Then forevery 0 < t < c , the polar body ( K c − ( t, ◦ is again a triangle given by( K c − ( t, ◦ = conv (cid:26)(cid:18) c − t , c c − t (cid:19) , (cid:18) c − t , − c c − t (cid:19) , (cid:18) − t , (cid:19)(cid:27) . Thus, a short calculation yields the volume formula, | ( K c − ( t, ◦ | = c t ( c − t ) . (4.3)Due to the axial symmetry of K c , its Steiner point s ( K c ) lies on the x -axis and itscoordinates are easily calculated to s ( K c ) = (cid:16) cπ arctan c , (cid:17) . | K c | | ( K c − s ( K c )) ◦ | = π arctan( c )(1 − π arctan( c )) , which tends to infinity as c tends to zero (by a computation involving onlyelementary calculus).For arbitrary n ≥
3, we consider the body of revolution L c ∈ K n , obtained byrotating the body K c around the e -axis of R n . The volume of L c can be easilycalculated and is given by | L c | = | B n − | nc n − , (4.4)where | B n − | denotes the ( n − R n − .Since for every K ∈ K n containing the origin in its interior and any subspace H ⊆ R n , we have K ◦ | H = ( K ∩ H ) ◦ , where the polar body on the right handside is taken in the subspace H , it follows, by taking H a 2-dimensional subspacecontaining e , that for every 0 < t < c , ( L c − te ) ◦ is a body of revolution obtainedby rotating the triangle ( K c − te ) ◦ around the e axis. Consequently, we obtainthe volume formula, | ( L c − te ) ◦ | = | B n − | n c n − t ( c − t ) n and, from this and (4.4), | L c || ( L c − te ) ◦ | = (cid:18) | B n − | n (cid:19) c n +1 t ( c − t ) n . Letting t = cg ( c ), where g ( c ) depends only on c and satisfies 0 < g ( c ) <
1, thisreduces to | L c || ( L c − cg ( c ) e ) ◦ | = (cid:18) | B n − | n (cid:19) g ( c )(1 − g ( c )) n which clearly tends to infinity if g ( c ) tends to zero as c tends to zero. It remainsto be shown that the e coordinate of the Steiner point of L c is of the form cg ( c )such that lim c → g ( c ) = 0 (note that by the rotational symmetry of L c all othercoordinates of s ( L c ) are zero). Since h ( L c , u ) = h ( K c , ( u · e ) e + p − ( u · e ) e )for every u ∈ S n − , we obtain from (2.5) by integration in cylinder coordinates(cf. [ , Lemma 1.3.1]), s ( L c ) · e = 1 | B n | Z S n − h ( L c , u )( u · e ) du = n Z − h ( K c , ζ e + p − ζ e ) ζ (1 − ζ ) n − dζ . K c , we see that s ( L c ) · e = n Z √ c − ζc (1 − ζ ) n − dζ + Z √ c cζ (1 − ζ ) n − dζ = c n − c n ( n + 1)(1 + c ) n +12 + Z √ c ζ (1 − ζ ) n − dζ | {z } = g ( c ) which is of the desired form. (cid:4) Theorems 1 and 4.1 raise the interesting problem whether there exist weaklymonotone or even non-monotonic Minkowski endomorphisms Φ (different frommultiples of J and − J) such that their volume product | K || Φ ◦ K | is either unboundedor maximized by convex bodies of non-empty interior (which we do not expect tonecessarily be Euclidean balls). Partial answers to this question were very recentlyobtained in [ ] by methods different to the ones used in this article.We turn now to inequalities for log-concave functions and begin with a functionalanalogue of Theorem 2 from which we will subsequently deduce Theorem 4. Theorem 4.2
Let µ be an SO( n − invariant probability measure on S n − withcenter of mass at the origin. If f ∈ LC c ( R n ) such that R R n f dx > , then Z R n (Ψ σ f ) ◦ ( x ) dx ≤ Z R n (Ψ µ f ) ◦ ( x ) dx ≤ Z R n f ◦ ( x ) dx. (4.5) There is equality in the left hand inequality if and only if Ψ µ f is radially symmetric.Equality in the right hand inequality holds if and only if f is even and Ψ µ = ∆ ⋆ orif f is radially symmetric.Proof. First note that for f ∈ LC c ( R n ), we always have R R n f dx < ∞ , by (2.7). Inorder to establish the left hand inequality of (4.5), we use (1.7), polar coordinatesand Jensen’s inequality to obtain Z R n (Ψ µ f ) ◦ ( x ) dx = n | B n | Z ∞ Z S n − exp( − ( h ( f, · ) ⊛ µ )( ru )) r n − dσ ( u ) dr ≥ n | B n | Z ∞ exp (cid:18) − Z S n − ( h ( f, · ) ⊛ µ )( ru ) dσ ( u ) (cid:19) r n − dr. (4.6)To be precise, for the application of Jensen’s inequality, we require the function( h ( f, · ) ⊛ µ )( r · ) to be σ -integrable. However, if this is not the case, then its integralis + ∞ and inequality (4.6) still holds. 23rom an application of (3.10) to the inner integral in (4.6) and the SO( n )invariance of σ , we conclude that Z R n (Ψ µ f ) ◦ ( x ) dx ≥ n | B n | Z ∞ exp (cid:18) − Z S n − h ( f, ru ) dσ ( u ) (cid:19) r n − dr = n | B n | Z ∞ exp( − ( h ( f, · ) ⊛ σ )( rv )) r n − dr for an arbitrary v ∈ S n − . Finally, using that ( h ( f, · ) ⊛ σ )( rv ) does not depend on v , we arrive at the left hand inequality of (4.5), Z R n (Ψ µ f ) ◦ ( x ) dx ≥ n | B n | Z ∞ Z S n − exp( − ( h ( f, · ) ⊛ σ )( rv )) r n − dσ ( v ) dr = Z R n (Ψ σ f ) ◦ ( x ) dx. If equality holds in the left hand inequality of (4.5), then we must have equality in(4.6) which implies by the equality condition of Jensen’s inequality (including thecase of non- σ -integrability) that for almost every r > c r ∈ ( −∞ , ∞ ]such that h (Ψ µ f, rv ) = ( h ( f, · ) ⊛ µ )( rv ) = c r for σ -a.e. v ∈ S n − . (4.7)Note that, by continuity, (4.7) yields that h (Ψ µ f, · ) is constant on every spherecontained in int dom h (Ψ µ f, · ). Next, we want to show that this domain is a ball.If c r < ∞ for some r >
0, then the lower semi-continuity of h (Ψ µ f, · ) impliesthat h (Ψ µ f, rv ) is finite for every v ∈ S n − . In particular, r S n − ⊆ dom h (Ψ µ f, · )which by the convexity of this domain yields rB n ⊆ dom h (Ψ µ f, · ). This impliesthat (4.7) holds for every r ′ ≤ r , by continuity, and c r ′ < ∞ . Thus, the set of all r > c r < ∞ is an interval, that is, there exists R > c r (cid:26) < ∞ for all r < R, = ∞ for all r > R for which (4.7) holds.In order to conclude that int dom h (Ψ µ f, · ) is a ball, it remains to show that for every r > R , h (Ψ µ f, · ) is infinite on r S n − . To this end, let x ∈ r S n − and assume that h (Ψ µ f, x ) < ∞ . Since dom h (Ψ µ f, · ) is convex and contains an open ball centered atthe origin, the convex hull of x and this ball is contained in dom h (Ψ µ f, · ). However,this convex hull must contain an open neighborhood of r ′ S n − for some r > r ′ > R for which (4.7) holds, which contradicts c r ′ = ∞ .Finally, since int dom h (Ψ µ f, · ) is a ball, by the comment following (4.7), h (Ψ µ f, · ) is radially symmetric on this ball. As a convex function which is radiallysymmetric (and, thus, depends only on one variable) on the interior of its domainmust be radially symmetric everywhere, we conclude that h (Ψ µ f, · ) is radiallysymmetric on R n . This concludes the proof of the equality conditions for the lefthand inequality of (4.5). 24or the proof of the right hand inequality of (4.5), we use (1.7), (2.16), andJensen’s inequality (note that µ is a probability measure) to obtain Z R n (Ψ µ f ) ◦ ( x ) dx ≤ Z R n Z S n − exp ( − h ( f, k x k ϑ x v )) dµ ( v ) dx. (4.8)As in the first part of this proof, for the application of Jensen’s inequality, we require h ( f, k x k ϑ x · ) to be µ -integrable. However, if this is not the case, then the left handside of (4.8) is zero and inequality (4.8) still holds.Since Ψ µ and the polar map commute with SO( n ) transforms, we may replace f in (4.8) by a rotated copy θf and integrate over SO( n ) with respect to the Haarmeasure, to arrive at Z R n (Ψ µ f ) ◦ ( x ) dx ≤ Z SO( n ) Z R n Z S n − exp (cid:0) − h ( f, k x k θ − ϑ x v ) (cid:1) dµ ( v ) dx dθ. (4.9)Since we integrate non-negative functions, we may apply Fubini’s theorem twice,the invariance of the Haar measure on SO( n ), and the fact that µ ( S n − ) = 1, toobtain the desired inequality Z R n (Ψ µ f ) ◦ ( x ) dx ≤ Z R n Z S n − Z SO( n ) exp (cid:0) − h ( f, k x k θ − ϑ x v ) (cid:1) dθ dµ ( v ) dx = Z R n Z SO( n ) exp (cid:0) − h ( f, θ − x ) (cid:1) dθ dx = Z SO( n ) Z R n f ◦ ( θ − x ) dx dθ = Z R n f ◦ ( x ) dx. By the above arguments, equality holds in the right hand inequality of (4.5)if and only if we have equality in (4.9) which implies by the equality condition ofJensen’s inequality (including the case of non- µ -integrability) that for almost every θ ∈ SO( n ) and almost every x ∈ R n there exist constants c θ,x ∈ ( −∞ , ∞ ] such that h ( f, k x k θ − ϑ x v ) = c θ,x for µ -a.e. v ∈ S n − . (4.10)As in the proof of Theorem 2, if µ is discrete it must coincide with the measure ν given by (3.3). Thus, Ψ µ = ∆ ⋆ and, since supp ν = {− ¯ e, ¯ e } , (4.10) reduces to theexistence of constants c θ,x ∈ ( −∞ , ∞ ] such that for almost every θ ∈ SO( n ) andalmost every x ∈ R n , h ( f, θ − x ) = c θ,x = h ( f, − θ − x ) . Consequently, the interior of the domain of h ( f, · ) must be origin-symmetric and h ( f, · ) must be even on it (by continuity, h ( f, · ) must attain the same value on all antipodal points in int dom h ( f, · )). By now considering the restriction of h ( f, · ) tolines through the origin and using the extendibility of convex, lower semi-continuousfunctions of one variable, we conclude that h ( f, · ) must be even on all of R n .25f µ is not discrete, we first want to show that (4.10) implies that int dom h ( f, · )is an open ball centered at the origin. To this end, note that it follows from (4.10)that for all r from a dense subset of (0 , ∞ ) and almost every η ∈ SO( n ), there existconstants c r,η ∈ ( −∞ , ∞ ] such that h ( f, rηv ) = c r,η for µ -a.e. v ∈ S n − . (4.11)If c r,η < ∞ for some r > η ∈ SO( n ), then the lower semi-continuity of h ( f, · ) implies that h ( f, rηv ) ≤ c r,η < ∞ for all v ∈ supp µ . If on the other hand c r,η = ∞ , then the lower semi-continuity of h ( f, · ) implies that v h ( f, rηv ) cannotbe bounded on any open subset of S n − intersecting supp µ .For suitable δ >
0, let B δ denote an open origin-symmetric δ -ball in dom h ( f, · )such that its closure is still contained in int dom h ( f, · ). Next, choose an arbitrary x ∈ int dom h ( f, · ) \ cl B δ and let r ∈ [ δ, k x k ). Then the set conv { x, cl B δ } iscontained in int dom h ( f, · ) and, thus, h ( f, · ) is bounded on it. Define the openspherical caps C rx as conv { x, B δ } ∩ r S n − .In the following, let d denote the geodesic distance on S n − and let d r denotethe geodesic distance on r S n − normalized such that d r ( u, v ) = d ( ur , vr ) for any u, v ∈ r S n − . Since µ is not discrete and has center of mass at the origin, thereexists t ∈ [0 ,
1) such that H ¯ e,t ∩ S n − ⊆ supp µ , where H ¯ e,t = { y ∈ R n : ¯ e · y = t } .Let α be the maximal geodesic distance of two points in H ¯ e,t ∩ S n − .Choose now r from the dense subset of all r ∈ [ δ, k x k ) such that (4.11) holdsfor almost all η ∈ SO( n ). Then for x ∈ r S n − with d r ( x , C r x ) < α and ε > A x ,ε = { η ∈ SO( n ) : r ηv ∈ C r x , d r ( r ηv , x ) < ε for some v , v ∈ H ¯ e,t ∩ S n − } . Clearly, A x ,ε is open and non-empty, hence, there exists η ∈ A x ,ε such that(4.11) holds for η and r . Since C r x is open and h ( f, · ) is bounded on C r x , wehave c r ,η < ∞ (as we have seen above). In particular, h ( f, r η v ) < ∞ , that is, r η v ∈ dom h ( f, · ) for some v ∈ H ¯ e,t ∩ S n − such that d r ( r η v , x ) < ε andthere exists v ∈ H ¯ e,t ∩ S n − with r η v ∈ C r x .Since x ∈ r S n − such that d r ( x , C r x ) < α and ε > h ( f, · ) is finite on a dense subset of U α/ ( C r x ), the set of all points on r S n − whose distance d r to C r x is less than α . Taking any r ′ < r , this implies that U α/ ( C r ′ x ) ⊆ dom h ( f, · ). Indeed, for every y ∈ U α/ ( C r ′ x ) we may find points x , . . . , x n in U α/ ( C r x ) ∩ dom h ( f, · ) such that y ∈ conv { , x , . . . , x n } ⊆ dom h ( f, · ).Since r can be chosen from a dense subset of [ δ, k x k ), we have shown that forevery x ∈ int dom h ( f, · ) and every r ∈ [ δ, k x k ), the set U α/ ( C rx ) is contained indom h ( f, · ). In particular, U α/ ( r k x k x ) ⊆ dom h ( f, · ).Finally, if int dom h ( f, · ) is not a centered open ball, then there exists a sequence x k ∈ int dom h ( f, · ) converging to x ∈ bd dom h ( f, · ) with k x k k > k x k > k (take, e.g., for x any boundary point that is touched non-radially by a closed ballin int dom h ( f, · )). 26ince x k → x , k x kk x k k x k converges to x as well. Hence, there exists k ∈ N suchthat d k x k (cid:16) k x kk x k k x k , x (cid:17) < α , that is, x ∈ U α/ (cid:18) k x kk x k k x k (cid:19) ⊆ [ r ∈ ( δ, k x k k ) U α/ (cid:18) r k x k k x k (cid:19) ⊆ int dom h ( f, · )which is a contradiction.Knowing that int dom h ( f, · ) is a centered open ball, (4.11) combined with thecontinuity of h ( f, · ) on the interior of its domain, implies, as in the final paragraphof the proof of Theorem 2, that h ( f, · ) is radially symmetric on int dom h ( f, · ).Noting again that a convex function which is radially symmetric on the interior ofits domain must be radially symmetric everywhere, we infer that h ( f, · ) is radiallysymmetric on all of R n . Since the Legendre transform commutes with the action ofSO( n ), f must be radially symmetric itself. As Ψ µ f = f for any radially symmetric f ∈ LC c ( R n ), by (2.16), this concludes the proof of the theorem. (cid:4) The same way Theorem 1 was a simple consequence of Theorem 2 and (1.1), wecan now deduce Theorem 4 easily from Theorems 4.2 and 2.6.
Proof of Theorem 4.
By the translation-invariance of Ψ µ , we have Ψ µ f = Ψ µ ˜ f ,where, as in Theorem 2.6, ˜ f ( x ) = f ( x − cent f ). Thus, by the right hand inequalityof (4.5) and Theorem 2.6, it follows that Z R n f ( x ) dx Z R n (Ψ µ f ) ◦ ( x ) dx = Z R n f ( x ) dx Z R n (Ψ µ ˜ f ) ◦ ( x ) dx ≤ Z R n f ( x ) dx Z R n ˜ f ◦ ( x ) dx ≤ (2 π ) n . Equality holds in (1.10) if and only if equality holds both in the right hand inequalityof (4.5) and in Theorem 2.6, that is, if and only if Ψ µ = ∆ ⋆ and f is a Gaussian orif f is proportional to a translation of the standard Gaussian. (cid:4) Let us remark at this point again that Theorem 1 can be recovered fromTheorem 4 in an asymptotically optimal form. More precisely, choosing f = K for K ∈ K n with non-empty interior in Theorem 4, inequality (1.10) becomes,(2 π ) n ≥ | K | Z R n ◦ Φ µ K ( x ) dx = | K | Z R n exp (cid:0) −k x k Φ ◦ µ K (cid:1) dx = n ! | K || Φ ◦ µ K | , where we have used that Ψ µ K = Φ µ K , the definition of the polar map, and (2.11)with p = 1. Since the assumption that µ is a probability measure is equivalent tothe normalization Φ µ B n = B n , we obtain | K || Φ ◦ µ K | ≤ (2 π ) n n ! = c nn | B n | , where c n > n →∞ c n = 1. 27he reason we do not recover the sharp form of Theorem 1 is the same reason,Urysohn’s inequality (1.2) is not a special case of its functional analogue (1.9),namely, that extremizers in the functional inequalities are Gaussians in both caseswhile the relevant geometric quantities are recovered for indicators of convex bodies.However, let us emphasize that the weakest inequality of Theorem 4, obtained for µ = σ , yields a new functional analogue of Urysohn’s inequality from which (1.2)can be deduced in an asymptotically optimal way, in contrast to (1.9). We willmake this even more precise with our final result which shows that all inequalitiesof Theorem 4 are strictly stronger than (1.9). The proof uses ideas from [ ]and relies on a basic inequality from information theory, sometimes attributed toShannon (see [ , Theorem B.1]), which states that if g, h : R n → R are non-negative measurable functions such that g > R R n g dx = 1, then Z R n g log 1 h dx ≥ Z R n g log 1 g dx − log (cid:18)Z R n h dx (cid:19) (4.12)with equality if and only if h = αg for some α ≥ Theorem 4.3
Let µ be an SO( n − invariant probability measure on S n − withcenter of mass at the origin. If f ∈ LC c ( R n ) , then n Z R n h ( f, x ) dγ n ( x ) ≥ − n log (cid:18) π ) n/ Z R n (Ψ µ f ) ◦ ( x ) dx (cid:19) (4.13) with equality if and only if Ψ µ f is a multiple of the standard Gaussian.Proof. First note that, by Lemma 3.4,2 n Z R n h ( f, x ) dγ n ( x ) = 2 n Z R n h (Ψ µ f, x ) dγ n ( x ) = 2 n Z R n log (cid:18) e − h (Ψ µ f,x ) (cid:19) ψ n ( x ) dx. Choosing g = ψ n and h = e − h (Ψ µ f, · ) in (4.12), we thus obtain2 n Z R n h ( f, x ) dγ n ( x ) ≥ n Z R n ψ n ( x ) log (cid:18) ψ n ( x ) (cid:19) dx − n log (cid:18)Z R n e − h (Ψ µ f,x ) dx (cid:19) . (4.14)The first integral on the right hand side is the entropy of the standard normaldistribution which is well known to be n (1 + log(2 π )). Consequently, we obtain2 n Z R n h ( f, x ) dγ n ( x ) ≥ π ) − n log (cid:18)Z R n (Ψ µ f ) ◦ ( x ) dx (cid:19) which is clearly equivalent to (4.13). Equality holds in (4.13) if and only if wehave equality in (4.14), that is, by the equality conditions of (4.12), if and only if e − h (Ψ µ f, · ) = αψ n , for some α > L ( − log Ψ µ f )( x ) = h (Ψ µ f, x ) = k x k β, for some β ∈ R and every x ∈ R n . This shows, by Proposition 2.4 (i) and (iv), thatequality holds in (4.13) if and only if Ψ µ f = e − β (2 π ) n/ ψ n . (cid:4) . Appendix The purpose of this appendix is to complete the proof of Proposition 3.3 byshowing that for ϕ ∈ Cvx ( o ) ( R n ), the function ϕ ⊛ µ is convex. The proof is basedon arguments used in [ ] and [ ], where variants of this fact were shown underadditional assumptions on the function ϕ . We begin with an auxiliary result. Lemma 5.1
Let ϕ : R n → R be convex and H ⊆ R n a -dimensional linearsubspace. For every z ∈ R n , a, b ∈ R , the function g z,a,b : H → R , defined by g z,a,b ( x ) = ϕ ( ax + bϑ H x + k x k z ) + ϕ ( ax + bϑ H x − k x k z ) , (5.1) where ϑ H ∈ SO( n ) acts as rotation by the angle π on H and keeps H ⊥ fixed, isconvex.Proof. Since ϕ and, thus, g z,a,b are continuous, it is sufficient to show that g z,a,b (cid:18) x + y (cid:19) ≤ g z,a,b ( x ) + 12 g z,a,b ( y ) (5.2)for all distinct x, y ∈ H . As, by definition, g z,a,b (cid:0) x + y (cid:1) equals ϕ (cid:18) a x + y bϑ H x + y k x + y k z (cid:19) + ϕ (cid:18) a x + y bϑ H x + y − k x + y k z (cid:19) , (5.3)we may only consider the first term for the following computation and then flip thesign of z . In order to see (5.2), first note that for every α ∈ [0 , a x + y bϑ H x + y k x + y k z = (cid:18) a x bϑ H x α k x + y k z (cid:19) + (cid:18) a y bϑ H y − α ) k x + y k z (cid:19) . Again, we may consider only the first term and skip the computation for the second(just replace x by y ). Choosing α = k x kk x k + k y k , we have 1 − α = k y kk x k + k y k and a x bϑ H x α k x + y k z = λ ( ax + bϑ H x + k x k z ) + λ ( ax + bϑ H x − k x k z ) , where λ = 14 (cid:18) k x + y kk x k + k y k (cid:19) and λ = 14 (cid:18) − k x + y kk x k + k y k (cid:19) . a x + y bϑ H x + y k x + y k z = λ ( ax + bϑ H x + k x k z ) + λ ( ax + bϑ H x − k x k z )+ λ ( ay + bϑ H y + k y k z ) + λ ( ay + bϑ H y − k y k z ) . Noting that λ , λ ∈ (cid:2) , (cid:3) and that 2 λ + 2 λ = 1, the convexity of ϕ implies that ϕ (cid:18) a x + y bϑ H x + y k x + y k z (cid:19) ≤ λ ϕ ( ax + bϑ H x + k x k z ) + λ ϕ ( ax + bϑ H x − k x k z )+ λ ϕ ( ay + bϑ H y + k y k z ) + λ ϕ ( ay + bϑ H y − k y k z ) . The analogue computation for the second term of (5.3), finally yields the desiredinequality (5.2). (cid:4)
In order to show that for ϕ ∈ Cvx ( o ) ( R n ), the function ϕ ⊛ µ is convex, wefirst assume that ϕ is convex and finite on R n . Moreover, we may restrict ourselvesto convex combinations along lines that lie completely in R n \{ } , the general casethen follows by continuity.Since a zonal function on S n − depends only on the value of u · ¯ e , there is anatural one-to-one correspondence between zonal functions and measures on S n − and functions and measures on [ − ,
1] (see, e.g., [ ]). In particular, there exists aunique non-negative measure b µ on [ − ,
1] such that for every f ∈ C ( S n − ), we have Z S n − f ( v ) dµ ( v ) = Z − Z S n − ∩ ¯ e ⊥ f (cid:16) α ¯ e + √ − α w (cid:17) dσ ¯ e ⊥ ( w ) (1 − α ) n − d b µ ( α ) , where σ ¯ e ⊥ is the invariant probability measure on S n − ∩ ¯ e ⊥ . Applying this todefinition (2.16), we obtain for x ∈ R n \{ } ,( ϕ ⊛ µ )( x ) = Z − Z S n − ∩ ¯ e ⊥ ϕ (cid:16) αx + √ − α k x k ϑ x w (cid:17) dσ ¯ e ⊥ ( w ) (1 − α ) n − d b µ ( α )= Z − Z S n − ∩ x ⊥ ϕ (cid:16) αx + √ − α k x k v (cid:17) dσ x ⊥ ( v ) (1 − α ) n − d b µ ( α ) . Since b µ is non-negative, we are done if we can prove the convexity of the function ϕ α ( x ) = Z S n − ∩ x ⊥ ϕ (cid:16) αx + √ − α k x k v (cid:17) dσ x ⊥ ( v ) , for all α ∈ [ − , x ∈ R n \{ } and let H ⊆ R n be an arbitrary2-dimensional linear subspace containing x . Then x ⊥ = H ⊥ ⊕ span { ϑ H x } .30irst, consider the case n = 3, where H = w ⊥ for some non-zero w ∈ R . Usingcylinder coordinates v = βϑ H x k x k ± p − β w on S ∩ x ⊥ , we obtain ϕ α ( x ) = 1 π Z − ϕ (cid:16) αx + √ − α βϑ H x + p (1 − α )(1 − β ) k x k w (cid:17) dβ p − β + 1 π Z − ϕ (cid:16) αx + √ − α βϑ H x − p (1 − α )(1 − β ) k x k w (cid:17) dβ p − β = 1 π Z − g z,a,b ( x ) dβ p − β , where z = p (1 − α )(1 − β ) w , a = α and b = √ − α β . By Lemma 5.1, g z,a,b isconvex and, hence, ϕ α is convex, as well.For n ≥
4, we again use cylinder coordinates on S n − ∩ x ⊥ in the direction of ϑ H x k x k to obtain ϕ α ( x ) = c n Z − Z S n − ∩ H ⊥ ϕ ( αx + √ − α βϑ H x + p (1 − α )(1 − β ) k x k w ) dσ H ⊥ ( v ) dβ (1 − β ) − n , where c n = n − ) √ π Γ( n − . Taking of this integral twice and replacing v by − v in onecopy, we see that again ϕ α ( x ) = c n Z − Z S n − ∩ H ⊥ g z,a,b ( x ) dσ H ⊥ ( v ) dβ (1 − β ) − n , where z = p (1 − α )(1 − β ) w , a = α and b = √ − α β as before. By Lemma 5.1, g z,a,b is convex and, thus, so is ϕ α .For general ϕ ∈ Cvx ( o ) ( R n ), we use that the Moreau envelope e t ϕ of ϕ is convexand finite and converges monotonously to ϕ , by Lemma 2.2. By what we haveshown above, each of the functions e t ϕ ⊛ µ , t >
0, is convex. Hence, by monotoneconvergence, we conclude that ϕ ⊛ µ = lim t ց e t ϕ ⊛ µ = sup t> e t ϕ ⊛ µ is convex as well. Acknowledgments
The authors were supported by the Austrian Science Fund(FWF), Project number: P31448-N35.
References [1] J. Abardia-Ev´equoz, A. Colesanti, and E. Saor´ın-G´omez,
Minkowski additive operators undervolume constraints , J. Geom. Anal. (2018), 2422–2455.
2] S. Alesker,
Valuations on convex functions and convex sets and Monge-Amp´ere operators ,Adv. Geom. (2019), 313–322.[3] M. Alexander, M. Fradelizi, and A. Zvavitch, Polytopes of maximal volume product , DiscreteComput. Geom. (2019), 583–600.[4] S. Artstein-Avidan and V. Milman, The concept of duality in convex analysis, and thecharacterization of the Legendre transform , Ann. of Math. (2009), 661–674.[5] S. Artstein-Avidan and V. Milman,
A characterization of the support map , Adv. Math. (2010), 379–391.[6] S. Artstein-Avidan and B.A. Slomka,
A note on Santal´o inequality for the polarity transformand its reverse , Proc. Amer. Math. Soc. (2015), 1693–1704.[7] S. Artstein-Avidan, B. Klartag, and V. Milman,
The Santal´o point of a function, and afunctional form of the Santal´o inequality , Mathematika (2004), 33–48.[8] K. Ball, Isometric problems in l p and sections of convex sets , Ph.D. Dissertation, TrinityCollege, Cambridge (1986).[9] F. Barthe, K.J. B¨or¨oczky, and M. Fradelizi, Stability of the functional forms of the Blaschke–Santal´o inequality , Monatsh. Math. (2014), 135–159.[10] A. Berg and F.E. Schuster,
Lutwak–Petty projection inequalities for Minkowski valuationsand their duals , J. Math. Anal. Appl. (2020), 124190, 24 pp.[11] K.J. B¨or¨oczky,
Stability of the Blaschke–Santal´o and the affine isoperimetric inequality , Adv.Math. (2010), 1914–1928.[12] L. Cavallina and A. Colesanti,
Monotone valuations on the space of convex functions , Anal.Geom. Metr. Spaces (2015), 167–211.[13] A. Colesanti, Functional inequalities related to the Rogers–Shephard inequality , Mathematika (2006), 81–101.[14] A. Colesanti, Log-concave functions , Convexity and concentration, 487–524, IMA Vol. Math.Appl. , Springer, New York, 2017.[15] A. Colesanti and I. Fragal´a,
The first variation of the total mass of log-concave functionsand related inequalities , Adv. Math. (2013), 708–749.[16] A. Colesanti, M. Ludwig, and F. Mussnig,
Minkowski valuations on convex functions , Calc.Var. Partial Differential Equations (2017), Art. 162, 29 pp.[17] A. Colesanti, M. Ludwig, and F. Mussnig, Valuations on convex functions , Int. Math. Res.Not. IMRN (2019), 2384–2410.[18] A. Colesanti, M. Ludwig, and F. Mussnig,
Hessian valuations , Indiana Univ. Math. J. (2020), 1275–1315.[19] A. Colesanti, M. Ludwig, and F. Mussnig, A homogeneous decomposition theorem forvaluations on convex functions , J. Funct. Anal. (2020), 108573, 25 pp.[20] D. Cordero-Erausquin, M. Fradelizi, G. Paouris, and P. Pivovarov,
Volume of the polar ofrandom sets and shadow systems , Math. Ann. (2015), 1305–1325.[21] F. Dorrek,
Minkowski endomorphisms , Geom. Funct. Anal. (2017), 466–488.[22] M. Fradelizi and M. Meyer, Some functional forms of Blaschke-Santal´o inequality , Math. Z. (2007), 379–395.[23] R.J. Gardner,
Geometric tomography , Second ed., Encyclopedia of Mathematics and itsApplications , Cambridge University Press, Cambridge, 2006.[24] R.J. Gardner and M. Kiderlen, Operations between functions , Comm. Anal. Geom. (2018),787–855.[25] H. Groemer, Geometric applications of Fourier series and spherical harmonics , Encyclopediaof Mathematics and its Applications , Cambridge University Press, Cambridge, 1996.
26] C. Haberl and F.E. Schuster,
Affine vs. Euclidean isoperimetric inequalities , Adv. Math. (2019), 106811, 26 pp.[27] J. Haddad, C.H. Jim´enez, and M. Montenegro,
Asymmetric Blaschke–Santal´o functionalinequalities , J. Funct. Anal. (2020), 108319, 18 pp.[28] G. Hofst¨atter, P. Kniefacz, and F.E. Schuster,
Affine Quermassintegrals and Minkowskivaluations , preprint.[29] M.N. Ivaki,
Convex bodies with pinched Mahler volume under the centro-affine normal flows ,Calc. Var. Partial Differential Equations (2015), 831–846.[30] B. Klartag and V.D. Milman, Geometry of log-concave functions and measures , Geom.Dedicata (2005), 169–182.[31] M. Kiderlen,
Blaschke- and Minkowski-endomorphisms of convex bodies , Trans. Amer. Math.Soc. (2006), 5539–5564.[32] J. Knoerr,
Smooth and mixed Hessian valuations on convex functions , preprint,arXiv:2006.12933.[33] J. Knoerr,
The support of dually epi-translation invariant valuations on convex functions ,preprint, arXiv:2005.00486.[34] A.V. Kolesnikov and E.M. Werner,
Blaschke–Santal´o inequality for many functions andgeodesic barycenters of measures , preprint, arXiv:2010.00135.[35] J. Lehec,
A direct proof of the functional Santal´o inequality , C. R. Math. Acad. Sci. Paris (2009), 55–58.[36] J. Lehec,
Partitions and functional Santal´o inequalities , Arch. Math. (2009), 89–94.[37] E. Lutwak, A general Bieberbach inequality , Math. Proc. Cambridge Philos. Soc. (1975),493–495.[38] E. Lutwak, Inequalities for mixed projection bodies , Trans. Amer. Math. Soc. (1993),901–916.[39] E. Lutwak and G. Zhang,
Blaschke–Santal´o inequalities , J. Differential Geom. (1997),1–16.[40] R. McEliece, The Theory of Information and Coding , Second ed., Encyclopedia ofMathematics and its Applications , Cambridge University Press, Cambridge, 2002.[41] M. Meyer and A. Pajor, On Santal´o’s inequality , Geometric Aspects of Functional Analysis(J. Lindenstrauss, V.D. Milman, eds), Lecture Notes in Math. , 261–263, Springer,Berlin, 1989.[42] M. Meyer and A. Pajor,
On the Blaschke–Santal´o inequality , Arch. Math. (1990), 82–93.[43] M. Meyer and S. Reisner, Shadow systems and volumes of polar convex bodies , Mathematika (2006), 129–148.[44] M. Meyer and E. Werner, The Santal´o-regions of a convex body , Trans. Amer. Math. Soc. (1998), 4569–4591.[45] R.T. Rockafellar,
Convex Analysis , Princeton Mathematical Series , Princeton UniversityPress, Princeton, N.J., 1970.[46] R.T. Rockafellar and R.J.B. Wets, Variational analysis , Fundamental Principles ofMathematical Sciences , Springer, Berlin, 1998.[47] L. Rotem,
On the mean width of log-concave functions , Geometric aspects of functionalanalysis, 355–372, Lecture Notes in Math. 2050, Springer, Heidelberg, 2012.[48] L. Rotem,
A sharp Blaschke–Santal´o inequality for α -concave functions , Geom. Dedicata (2014), 217–228.[49] R. Schneider, Bewegungs¨aquivariante, additive und stetige Transformationen konvexerBereiche , Arch. Math. (1974), 303–312.
50] R. Schneider,
Additive Transformationen konvexer K¨orper , Geometriae Dedicata (1974),221–228.[51] R. Schneider, Equivariant endomorphisms of the space of convex bodies , Trans. Amer. Math.Soc. (1974), 53–78.[52] R. Schneider,
Convex Bodies: The Brunn–Minkowski Theory , Second ed., Encyclopedia ofMathematics and its Applications , Cambridge University Press, Cambridge, 2013.[53] F.E. Schuster,
Convolutions and multiplier transformations of convex bodies , Trans. Amer.Math. Soc. (2007), 5567–5591.[54] F.E. Schuster,
Crofton Measures and Minkowski Valuations , Duke Math. J. (2010),1–30.[55] F.E. Schuster and T. Wannerer,