Rogers-Shephard inequality for log-concave functions

arXiv:1410.2556v2 [math.FA] 13 Sep 2016 ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS DAVID ALONSO-GUTI´ ERREZ, BERNARDO GONZ´ ALEZ MERINO, C. HUGO JIM´ ENEZ, RAFAEL VILLA Abstract. In this paper we prove different functional inequalities extending the classical Rogers-Shephard inequalities for convex bodies. The original inequalities provide an optimal relation between the volume of a convex body and the volume of several symmetrizations of the body, such as, its difference body. We characterize the equality cases in all these inequalities. Our method is based on the extension of the notion of a convolution body of two convex sets to any pair of log-concave functions and the study of some geometrical properties of these new sets. 1. Introduction A measure µon Rnis log-concave if for any measurable sets A, B ⊂Rnand 0< λ < 1, µ(λA + (1 −λ)B)≥µ(A)λµ(B)1−λ whenever A, B ⊂Rnand λA + (1 −λ)Bare measurable, where A+B={a+b: a∈A, b ∈B}is the Minkowski sum. Log-concave measures naturally appear in Convex Geometry, since the Brunn Minkowski inequality establishes the log-concavity of the Lebesgue measure restricted to convex sets, and of the marginal sections of convex sets. A function f:Rn→[0,+∞) is log-concave if f(x) = e−u(x)for some convex function u:Rn→(−∞,∞]. As was shown in [16], a measure µon Rnwith fulldimensional support is log-concave if and only if it has a log-concave density with respect to the Lebesgue measure. The class of log-concave functions has proven to be of great importance in several areas or mathematics. From a functional point of view it has been shown they resemble Gaussian functions in many different ways. Many functional inequalities satisfied by Gaussian functions, like Poincare and Log-Sobolev inequalities, also hold in a more general subclass of log-concave functions [8, 9, 12]. They also appear in areas as Information Theory, in the study of some important parameters, such as the classical entropy [14]. There are many examples in the literature of functional inequalities with a geometric counterpart; Prekopa-Leindler/Brunn-Minkowski [27] and Sobolev/Petty projection [34] inequalities are two of the main examples. This has generated an increasing interest to extend several important parameters of convex bodies to functional parameters [5, 6, 12, 19, 22, 24] in the class of logconcave functions. Date: September 14, 2016. 2010 Mathematics Subject Classification. Primary 52A20, Secondary 39B62,46N10. Key words and phrases. Rogers-Shephard inequality, log-concave measures, log-concave functions, convolution body, geometric inequalities. 1 2 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA The class of log-concave functions is often regarded as the natural extension of convex bodies, taking into account that the characteristic function of a convex body is a log-concave function and that this is the smallest closed under limits class of functions that contains the densities of n-dimensional marginals of uniform probabilities on convex bodies of higher dimension (we refer to the next section for precise definitions). In this work we extend Rogers-Shephard inequality [28, 29] to the class of logconcave functions (Theorems 2.1 and 2.3), characterizing the equality cases as well. We provide other functional versions of inequalities around Rogers-Shephard’s with their respective characterization of equality cases. While the Brunn-Minkowski inequality is commonly seen as the backbone of modern Convex Geometry, RogersShephard inequality can be considered as a reverse form of Brunn-Minkowski inequality that not only describes a relation between Minkowski addition and volume, but also deals with yet another fundamental property in convexity: symmetry. The far reaching influence of this inequality becomes evident as it can be found as an ingredient not only in many important works in classical and asymptotic convex geometry [11, 23, 21, 26] but also in many others with a more analytical flavor [31, 7, 33, 25] and its extension to a functional setting as well as for the entropy of convex measures has already been considered for instance in [17] or [15]. The paper is organised as follows: In Section 2 we provide the notation used in the rest of the paper and some previous results and state the precise results we are going to prove. In Section 3 we introduce the (θ, t)-convolution bodies of logconcave functions and prove the extension of Rogers-Shephard inequalities (8) and (9). In Section 4 we introduce the more general concept of k-th (θ, t)-convolution bodies and prove a Rogers-Shephard type inequality (10) for surface area. When particularizing to k=nwe obtain the previously introduced (θ, t)-convolution bodies. In Section 5 we characterize the equality cases in these inequalities. Finally, in Section 6, we revisit another result around Rogers-Shephard inequality [17] by giving an extended version for two different functions. Thus, we extend inequality (7) for any two log-concave functions and characterize the equality cases. Since this inequality will strengthen another well known Rogers-Shephard inequality (4) we will make use of these new tools to characterize the equality cases in inequality (4). 2. Notation and previous results The notation used in this paper is quite standard in modern convex geometry and consistent with for example [32] and [20]. A convex body is a subset of Rnthat is convex, compact and has non-empty interior. It is said to be centrally symmetric if for any x∈Kwe have that also −x∈K. When studying geometric properties of a convex body Kit is usually very convenient to construct another convex body from Kwhich is centrally symmetric. There are many ways to construct such a symmetrization. One of them is the so called difference body of K, which is the Minkowski sum of Kand −K. Let us recall that the Minkowski sum of two convex bodies Kand Lis defined as K+L={x+y∈Rn:x∈K, y ∈L} ={x∈Rn:K∩(x−L)6=∅}. Brunn-Minkowski inequality (see, for instance, [4] for several proofs and characterization of the equality cases) gives the following lower bound for the volume of ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 3 the Minkowski sum of any two convex bodies K, L ⊆Rn: |K+L|1 n≥ |K|1 n+|L|1 n. As a consequence of Brunn-Minkowski inequality the following relation between the volume of the difference body K−Kand the volume of Kis always true |K−K| ≥ 2n|K|, with equality if and only if Kis symmetric. In [28] Rogers and Shephard proved a reverse inequality. Namely, Rogers-Shephard inequality states that for any convex body K⊆Rn, we have (1) |K−K| ≤ 2n n|K|, with equality if and only if Kis a simplex. This inequality was extended to any pair of convex bodies K, L ⊆Rnin [29], showing that (2) max x0∈Rn|K∩(x0−L)||K+L| ≤ 2n n|K||L|, with equality if and only if K=−Lis a simplex (see [2] for the characterization of equality). In the same paper [29] the authors also considered different types of symmetrization of a convex body Kand proved volume inequalities for them. In particular it was shown that for any convex body K⊆Rncontaining 0, the volume of the convex hull of Kand −Kverifies (3) |conv{K, −K}| ≤ 2n|K| with equality if and only if Kis a simplex and 0 is one of its vertices. In the same paper the authors remarked that, with a similar proof, the latter inequality can be extended to the following inequality for any two bodies Kand Lcontaining the origin (4) |K∩L||conv{K, −L}|≤ 2n|K||L| and they suggested that it is likely that equality is attained if and only if K=Lis a simplex and 0 is one of its vertices. Very recently, in [2], the volume of the θ-convolution bodies K+θLwas studied, where (5) K+θL={x∈K+L:|K∩(x−L)| ≥ θmax z∈Rn|K∩(z−L)|}. As a consequence of the volume inequalities obtained for convolution bodies, inequality (2) was recovered and the equality cases were characterized. In [1], similar inclusion relations and volume inequalities were obtained for the h, θ-convolution bodies of Kand L, defined as K+h,θ L={x∈K+L:h(K∩(x−L)) ≥θmax z∈Rnh(K∩(z−L))}, where his a function satisfying some properties. As a particular case we have the k-th θ-convolution bodies of two convex bodies, defined as K+k,θ L={x∈K+L:Wn−k(K∩(x−L)) ≥θmax z∈RnWn−k(K∩(z−L))}, 4 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA where Wn−kdenotes the (n−k)-th quermaßintegral of a convex body which, by Kubota’s formula (cf. [32, p. 295]), can be expressed as an average of the volumes of the k-dimensional projections of K Wn−k(K) = |Bn 2| |Bk 2|ZGn,k |PE(K)|dµ(E). (Gn,k denotes the set of k-dimensional linear subspaces, dµ is the Haar probability measure on Gn,k and PE(K) is the projection of Kon a subspace E). As a consequence of these volume inequalities the following Rogers-Shephard type inequality for any two convex bodies K, L ⊆Rn, which involves the surface area of Kand L, is obtained (6) |K+L| ≤ 2n n|K||∂L|+|L||∂K| 2 maxx0∈Rn|∂(K∩(x0−L))|, where |∂K|is the surface area of K. Notice that when L=−Kwe recover inequality (1). Let us recall that, up to a constant which depends only on the dimension n, the surface area of a convex body Kequals the quermaßintegral W1(K). Inequality (3) was extended to the context of log-concave functions in [17], where the author proved that for any log-concave function f, if its difference function is defined by ∆f(z) = sup npf(x)f(−y) : x, y ∈Rn: 2z=x+yo, then (7) ZRn ∆f(x)dx ≤2nZRn f(x)dx. Taking f(x) = e−hK◦(x), with hK◦(x) = maxy∈K◦hx, yithe support function of the polar set of a convex body Kcontaining the origin, inequality (3) is recovered. This inequality was also extended in [3]. In this paper we extend inequalities (1), (2) and (6) to log-concave functions. Before we state our results we need to introduce some more notation. Given f, g two log-concave functions, their convolution defined by f∗g(x) = ZRn f(z)g(x−z)dz is also a log-concave function in Rn. If f(x) = χK(x) and g(x) = χL(x) are the characteristic functions of two convex bodies, then f∗g(x) = |K∩(x−L)|. The Asplund product of two log-concave functions is defined by f ⋆ g(x) = sup z∈Rn f(z)g(x−z). If f(x) = χK(x) and g(x) = χL(x), then f ⋆ g(x) = χK+L(x). This operation is the natural extension of the Minkowski sum of convex bodies, as it has been shown when extending geometric inequalities to the context of general log-concave functions (see for instance [5]). Remark. Notice that if both fand gare integrable and continuous when restricted to their supports then this supremum is a maximum since, in such case, if f ⋆g(x) = 0, then for any z∈Rnwe have that f(z)g(x−z) = 0 and if f ⋆ g(x)>0, then there exists a t > 0 such that the set At(x) := {z∈supp f∩(x−supp g) : f(z)g(x−z)≥tkfk∞kgk∞} ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 5 is not empty. Since fand gare integrable log-concave functions this set is convex and bounded. Thus, its closure is a compact convex set. Since both fand gare continuous when restricted to their supports and f ⋆ g(x) = sup z∈Rn f(z)g(x−z) = sup z∈cl(At(x)) f(z)g(x−z) the function f(z)g(x−z) is continuous on the compact set At(x) and the maximum is attained. With this notation, we prove the following extension of inequality (2). Theorem 2.1. Let f, g :Rn→Rbe two integrable log-concave functions with full-dimensional support such that fand gare continuous when restricted to their supports. Then (8) kf∗gk∞ZRn f ⋆ g(x)dx ≤2n nkfk∞kgk∞ZRn f(x)dx ZRn g(x)dx. Furthermore, this inequality becomes an equality if and only if f(x) kfk∞=g(−x) kgk∞is the characteristic function of an n-dimensional simplex. In case we consider g(x) = f(−x) the latter inequality can be improved to the following extension of inequality (1): Theorem 2.2. Let fbe a log-concave function with full-dimensional support and continuous when restricted to it and let ¯ f(x) = f(−x). Then (9) ZRn f ⋆ ¯ f(x)dx ≤2n nkfk∞ZRn f(x)dx. Furthermore, this inequality becomes an equality if and only if f(x) kfk∞is the characteristic function of an n-dimensional simplex. Let us mention that the previous inequality was first obtained by Colesanti [17, Theorem 4.3] where the author proves it in the quasi-concave case without characterizing the equality case. The fact that inequality (9) is an improvement of inequality (8) follows from Young’s inequality kf∗¯ fk∞≤ kfk1kfk∞. The notion of quermaßintegrals has also been extended from convex bodies to the setting of log-concave functions. In [18], [22] and [30], the case of the perimeter and the mean width is considered while, in [13], a different definition is given for all the quermaßintegrals. We will work with the definition in the latter paper, where, in particular, the quermaßintegral W1(surface area) of a log-concave function is defined by W1(f) := Z∞ 0 W1({x∈Rn:f(x)≥t})dt. By Crofton’s formula (cf. [32, p. 235]), this equals W1(f) = cnZAn,1 max z∈Ef(z)dµn,1(E), where cn=|Bn 2| |Bn−1 2|is a constant depending only on nand An,1is the set of affine 1-dimensional subspaces of Rnand µn,1is the Haar probability measure on it. 6 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA Theorem 2.3. Let f, g :Rn→Rbe two integrable log-concave functions with full-dimensional support and continuous when restricted to their supports. Then (10) ZRn f ⋆ g(x)dx ≤2n nkfk∞kgk∞ W1(g)RRnf(x)dx +W1(f)RRng(x)dx 2 maxx0∈RnW1(f(·)g(x0−·)) . Furthermore, when n≥3this inequality becomes an equality if and only if f(x) kfk∞= g(−x) kgk∞is the characteristic function of an n-dimensional simplex. Finally, we will prove the following extension of (7). Before stating it let us start with the following definition: Definition 2.1. Let f, g :Rn→Rbe two integrable log-concave functions. Define f⊕g(z) := sup 2z=x+ypf(x)g(y) = pf ⋆ g(2z). Following the proof given in [17] of inequality (7), we can show the following result. The inequality in the result was also obtained in [3] in the more general case where z=λx + (1 −λ)yand not just λ=1 2. However, equality cases need a more detailed argument. Theorem 2.4. Let f, g :Rn→Rbe two integrable log-concave functions with full-dimensional supports and continuous when restricted to them. Then (11) ZRnpf(x)¯g(x)dx ZRn f⊕g(x)dx ≤2nZRn f(x)dx ZRn g(x)dx. Equality holds if and only if the following two conditions are satisfied: (i) supp f=supp ¯gis a translation of a cone Cwith vertex at 0 with simplicial section, and (ii) f(x) = c1e−ha,xion supp fand g(x) = c2e−hb,xion supp gfor some c1, c2>0 and some a, b ∈Rnsuch that ha, xi ≥ 0≥ hb, xifor every x∈C. 3. (θ, t)-convolution bodies of log-concave functions and Rogers-Shephard inequalities In this section we prove the aforementioned extensions of Rogers-Shephard inequality to log-concave functions. In order to prove them we need to introduce some more notation. Given f, g two integrable log-concave functions with fulldimensional support, x∈supp f+ supp gand t∈(0,1], let us recall that we denote At(x) = At(f, g)(x) := {z∈supp f∩(x−supp g) : f(z)g(x−z)≥tkfk∞kgk∞}. Since fand gare integrable log-concave functions, At(x) is a bounded convex set. Definition 3.1. Let f, g :Rn→Rbe two integrable log-concave functions with full-dimensional support, t∈(0,1],θ∈[0,1]. We define the (θ, t)-convolution set of fand gas the set Cθ,t =Cθ,t(f, g) := {x∈supp f+supp g:At(x)6=∅,|At(x)| ≥ θMt} where Mt=Mt(f, g) := max x0∈supp f+supp g|At(x0)|. ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 7 Remark. When f(x) = χK(x) and g(x) = χL(x) are the characteristic functions of two convex bodies Kand L, the sets At(x) = K∩(x−L) for any t∈(0,1] and we recover the definition of the θ-convolution bodies K+θLin [2]. It is obvious from the definition that for any fixed t, the sets Cθ,t decrease on θ. The following lemma implies the convexity of these sets and gives a reverse (increasing on θ) inclusion relation when normalized by the right factor, as shows Corollary 3.2. Lemma 3.1. Let t∈(0,1],f, g :Rn→Rbe two integrable log-concave functions with full-dimensional support such that Mt=|At(0)|,θ1, θ2, λ1, λ2∈[0,1] with λ1+λ2≤1. Then λ1Cθ1,t +λ2Cθ2,t ⊆ Cθ,t, with 1−θ1 n=λ1(1 −θ 1 n 1) + λ2(1 −θ 1 n 2). Proof. Let x1∈ Cθ1,t,x2∈ Cθ2,t. For any z0∈ At(0), z1∈ At(x1), z2∈ At(x2), the log-concavity of fand gimplies f((1 −λ1−λ2)z0+λ1z1+λ2z2)g(λ1x1+λ2x2−(1 −λ1−λ2)z0−λ1z1−λ2z2) ≥(f(z0)g(−z0))1−λ1−λ2(f(z1)g(x1−z1))λ1(f(z2)g(x2−z2))λ2≥tkfk∞kgk∞. Thus, At(λ1x1+λ2x2)⊇(1 −λ1−λ2)At(0) + λ1At(x1) + λ2At(x2) and, by Brunn-Minkowski inequality |At(λ1x1+λ2x2)|1 n≥ (1 −λ1−λ2)|At(0)|1 n+λ1|At(x1)|1 n+λ2|At(x2)|1 n≥ (1 −λ1−λ2)M 1 n t+λ1θ 1 n 1M 1 n t+λ2θ 1 n 2M 1 n t= (1 −λ1(1 −θ 1 n 1)−λ2(1 −θ 1 n 2))M 1 n t. Consequently, λ1x1+λ2x2∈ Cθ,t. In particular, taking θ1=θ2and λ1+λ2= 1 we obtain that these sets Cθ,t are convex. Besides Corollary 3.2. Let t∈(0,1],f, g :Rn→Rbe two integrable log-concave functions with full-dimensional support such that Mt=|At(0)|,0≤θ0≤θ < 1. Then Cθ0,t 1−θ 1 n 0⊆Cθ,t 1−θ1 n . Proof. Taking θ1=θ2=θ0in the previous lemma, we have that for any λ1, λ2∈ [0,1] with λ1+λ2≤1 (λ1+λ2)Cθ0,t =λ1Cθ0,t +λ2Cθ0,t ⊆ Cθ,t, with (λ1+λ2)(1 −θ 1 n 0) = 1 −θ1 n. Thus, for any θ0≤θ≤1, taking λ1+λ2=1−θ1 n 1−θ 1 n 0 we obtain the result.  In a similar way we can prove the following: 8 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA Lemma 3.3. Let f, g :Rn→Rbe two integrable log-concave functions with fulldimensional support. Then for any t1, t2∈(0,1] and any λ∈[0,1] we have M 1 n tλ 1t1−λ 2≥λM 1 n t1+ (1 −λ)M 1 n t2. Consequently, Mtis continuous on (0,1). Proof. Let x1, x2be such that Mti=|Ati(xi)|for i= 1,2. Since fand gare log-concave we have that for any z1∈ At1(x1), z2∈ At2(x2) and any λ∈,[0,1] f(λz1+ (1 −λ)z2)g(λx1+ (1 −λ)x2−(λz1+ (1 −λ)z2)) ≥(f(z1)g(x1−z1))λ(f(z2)g(x2−z2))1−λ ≥tλ 1t1−λ 2kfk∞kgk∞. Thus, Atλ 1t1−λ 2(λx1+ (1 −λ)x2)⊇λAt1(x1) + (1 −λ)At2(x2). By Brunn-Minkowsi inequality M 1 n tλ 1t1−λ 2≥ |Atλ 1t1−λ 2(λx1+ (1 −λ)x2)|1 n≥λM 1 n t1+ (1 −λ)M 1 n t2. Consequently, the function f(s) = M 1 n esis concave in (−∞,0] and then it is continuous on (−∞,0). Thus, Mt=fn(log t) is continuous on (0,1).  Let us now prove inequality (8). Proof of Theorem 2.1 (inequality). We can assume, without loss of generality, that kfk∞=kgk∞= 1. By definition of Cθ,t, we have that for any t∈(0,1] C0,t ={x∈suppf+ suppg:At(x)6=∅} ={x∈suppf+ suppg:f ⋆ g(x)≥t}. For any t∈(0,1], let x0(t)∈Rnbe such that Mt=|At(x0)|. By Corollary 3.2 with θ0= 0 and greplaced by g(·+x0), for any θ∈[0,1] (1 −θ1 n)(−x0(t) + C0,t)⊆ −x0(t) + Cθ,t. Taking volumes and integrating in θ∈[0,1] we obtain |C0,t| ≤ 2n nZ1 0|Cθ,t|dθ =2n nZRn |At(x)| Mt dx. Consequently Mt|C0,t| ≤ 2n nZRn|At(x)|dx and, integrating in t∈(0,1] Z1 0 Mt|C0,t|dt ≤2n nZ1 0ZRn|At(x)|dxdt. The integral on the right-hand side is Z1 0ZRn|At(x)|dxdt =ZRnZRn f(z)g(x−z)dzdx =ZRn f(x)dx ZRn g(x)dx. ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 9 On the other hand, the integral on the left-hand side is Z1 0 Mt|C0,t|dt =ZRnZf⋆g(x) 0 max x0∈Rn|At(x0)|dtdx ≥max x0∈RnZRnZf⋆g(x) 0|At(x0)|dtdx = max x0∈RnZRnZRn min {f ⋆ g(x), f(z)g(x0−z)}dzdx. Since kfk∞=kgk∞= 1, both quantities in the minimum are smaller than or equal to 1, the minimum is bounded from below by the product and so, this quantity is bounded from below by max x0∈Rnf∗g(x0)ZRn f ⋆ g(x)dx. Thus kf∗gk∞ZRn f ⋆ g(x)dx ≤2n nkfk∞kgk∞ZRn f(x)dx ZRn g(x)dx.  Let us now consider the case in which g(x) = f(−x). We will denote this function ¯ fand At(f)(x) := At(f, ¯ f)(x). Notice that for any t∈(0,1) At(f)(0) = z∈Rn:f(z)2≥tkfk2 ∞ =nz∈Rn:f(z)≥√tkfk∞o. Analogously, let us denote Mt(f) := Mt(f, ¯ f) and Cθ,t(f) := Cθ,t(f, ¯ f). The following lemma shows that the maximum value of |At(f)(x)|is attained at x= 0. Lemma 3.4. Let f:Rn→Rbe an integrable log-concave function with fulldimensional support, then for any t∈(0,1), At(f)(x)⊆1 2x+At(f)(0). Consequently, Mt(f) = |At(f)(0)|. Proof. Since fis log-concave, for any x∈supp f−supp f At(f)(x) = (z∈Rn:sf(z) kfk∞ f(z−x) kfk∞≥√t) ⊆z∈Rn:f(z−1 2x) kfk∞≥√t =1 2x+At(f)(0).  The following lemma shows a relation between the (θ, t)-convolution bodies of f and ¯ fand the θ-convolution bodies of At(f)(0) and −At(f)(0). 16 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA Notice that taking θtending to 1 in (i) we have that for every t∈(0,1) the maximum of |At(x)|is only attained at x0(t). Besides, by the continuity of Mt, the continuity of |At(x0)|in tand in x0, (ii) holds if and only if x0(t) is the same for every t∈(0,1), and thus we may suppose without loss of generality that x0(t) = 0. Thus (i) for every θ, t ∈(0,1), (1 −θ1 n)C0,t =Cθ,t, (ii) for every t∈(0,1), maxx0∈Rn|At(x0)|=|At(0)| (iii) for every x, z ∈Rn, min{f ⋆ g(x), f(z)g(−z)}=f ⋆ g(x)f(z)g(−z). First of all notice that if g(x) = χK(x) is the characteristic function of a convex body, then At(x) = At2(f)(0) ∩(x−K) and Cθ,t =At2(f)(0) +θK, where At2(f)(0) +θKdenotes the θ-convolution of convex bodies defined in (5). Thus, since the (θ, t)-convolution bodies of the functions are the θ-convolution bodies of some convex bodies we have equality in (i) if and only if for every t∈(0,1) At2(f)(0) = −Kis a simplex and, consequently f(x) = g(−x) is the characteristic function of a simplex. We will prove that in the equality case necessarily one of the functions is the characteristic function of a convex body. Condition (iii) occurs if and only if f ⋆ g(x) or f(z)g(−z) equals 0 or 1, for every x, z ∈Rn. First, assume that f ⋆ g(x) = 1 for every x∈supp f+ supp g. Then for every x∈supp f+ supp g,A1(f)(0) ∩(x−A1(g)(0)) 6=∅and so A1(f)(0) + A1(g)(0) = supp f+ supp g. Consequently fand gare characteristic functions. Let us now assume that there exists x∈supp f+ supp gsuch that f ⋆ g(x)<1. Then for every z∈Rn,f(z)g(−z) equals 0,1 and then, for every t∈(0,1], At(0) = A1(0). In such case the function Mtis constant in (0,1]. In particular it is also continuous on t= 1 and then (i) also holds for t= 1. Notice that if |A1(0)|=|At(0)|= 0, then for every t∈(0,1) we have that for every θ∈(0,1) Cθ,t ={x∈suppf +suppg :At(x)6=∅}, contradicting (i). Thus, if we have equality in (8), |A1(0)|>0. Now, if (i) holds, we fix t∈(0,1] and take x∈supp(f) + supp(g). If x /∈ C0,t then At(x) = ∅. If x∈ C0,t then there exists θx∈[0,1] such that x∈∂Cθx,t and x= (1 −θ 1 n x)yfor some y∈∂C0,t. Thus, we have equality in θ 1 n x|At(0)|1 n=|At(x)|1 n=|At(θ 1 n x0 + (1 −θ 1 n x)y)|1 n ≥θ 1 n x|At(0)|1 n+ (1 −θ 1 n x)|At(y)|1 n≥θ 1 n x|At(0)|1 n. and, by the equality cases in Brunn-Minkowski inequality, At(x) is homothetic to At(0). Thus, for every x∈Rnand t∈(0,1], At(x) is either empty or a homothetic copy of At(0). Moreover, if we particularize in t= 1, we have that A1(x) ={z:f(z)g(x−z) = 1}={z:f(z) = 1}∩(x+{z:g(−z) = 1}) and A1(0) ={z:f(z)g(−z) = 1}={z:f(z) = 1}∩{z:g(−z) = 1} ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 17 and for any θ∈[0,1] then Cθ,1is the θ-convolution of the convex bodies Cθ,1=A1(f)(0) +θA1(g)(0). Thus, by Proposition 2.10 in [2], the convex bodies A1(f)(0) and −A1(g)(0) are the same simplex A1(0). Let us now assume that none of the functions f, g is a characteristic function, and we will find a contradiction. Since fand gare not characteristic functions there exist 0 < t1, t2<1 and z1, z2∈Rnsuch that t1≤f(z1)<1 and t2≤g(−z2)<1. Let us denote by F1a facet of A1(0), with outer normal vector u1and contained in the hyperplane {x∈ Rn:hx, u1i=c1}, such that A1(0) ⊆ {x∈Rn:hx, u1i ≤ c1}and hz1, u1i> c1. Analogously, let F2be a facet of A1(0), with outer normal vector u2and contained in the hyperplane {x∈Rn:hx, u2i=c2}, such that A1(0) ⊆ {x∈Rn:hx, u2i ≤ c2} and hz1, u2i> c2. Observe that the log-concavity of fand gimply that conv{z1,A1(0)} ⊂ At2 1(f)(0) and conv{z2,A1(0)} ⊂ At2 2(¯g)(0). If F1=F2, then (conv{z1,A1(0)}∩conv{z2,A1(0)})\A1(0) 6=∅. Let z0be a point in this intersection. Then z0∈ At1t2(0) since t1t2≤f(z0)g(−z0)<1. However, this is not possible, since At1t2(0) = A1(0). If F16=F2, let xbe a vector with a small enough norm and parallel to the only edge contained in all n−1 facets F3,...,Fn+1 of A1(0) different from F1and F2 and pointing from F2to F1such that z1/∈x+A1(0) and A1(0) ∩(x+F2)6=∅, (conv{z1,A1(0)}\{z1,A1(0)})∩(x+F1)6=∅, and A1(0) ∩(x+ conv{z2,A1(0)}\{z2,A1(0)})6=∅. Observe that for any point zin the first intersection, it holds t1t2< t1≤f(z)g(x− z), whereas if z′is on the second, then t1t2< t2≤f(z′)g(x−z′) and, since xpoints from F2to F1,z′does not belong to x+A1(0). Besides, for any other facet Fi∩(x+Fi)6=∅and for any point z′′ in Fi∩(x+Fi) we have f(z′′)g(x−z′′) = 1. These three types of points belong to the set At0t1(x), which we have proved that is a homothetic copy of the simplex At1t2(0) = A1(0). Then, since for every facet Fithere exist points in (x+Fi)∩At1t2(x) we have that x+A1(0) ⊆At1t2(x) and since the points z′∈At1t2(x)\(x+A1(0)) the inclusion is strict. Thus |At1t2(0)|=|A1(0)|=|x+A1(0)|<|At1t2(x)|, contradicting (ii). Thus, for gmust be a characteristic function and, consequently, f(x) = g(−x) is the characteristic function of a simplex.  The proof of the equality cases in (10) follows the same lines. Proof of Theorem 2.3 (Equality). Without loss of generality we assume that ||f||∞= ||g||∞= 1. 18 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA Then, equality holds in (10) if and only if it holds equality on each inequality all along the proof of (10). More particularly, if x0(t) is such that M1,t =W1(At(x0)) we have that (i) for every θ, t ∈(0,1), (1 −θ1 n−1)(−x0(t) + Cn−1 0,t ) = −x0(t) + Cn−1 θ,t , (ii) max x0∈RnZRnZf⋆g(x) 0 W1(At(x0))dtdx =ZRnZf⋆g(x) 0 max x0∈RnW1(At(x0(t)))dtdx (iii) for every x∈Rn, E ∈An,1, min{f ⋆ g(x),max z∈Ef(z)g(x0(t)−z)}=f ⋆ g(x) max z∈Ef(z)g(x0(t)−z) (iv) for every θ∈Sn−1and for every z, w ∈θ⊥,r, s ∈R, max s∈Rfz(s)gw(r−s) = max s∈Rmin{fz(s)kgwk, gw(r−s)kfzk∞}. Taking θtending to 1 in (i) we have that for every t∈(0,1] the maximum of W1(At(x)) is only attained at x0(t). Besides, (ii) holds if and only if x0(t) is the same for every t, and thus we may suppose without loss of generality that x0(t) = 0. Thus (i) for every θ, t ∈(0,1), (1 −θ1 n−1)Cn−1 0,t =Cn−1 θ,t , (ii) for every t∈(0,1], maxx0∈RnW1(At(x0)) = W1(At(0)) (iii) for every x∈Rn, E ∈An,1, min{f ⋆ g(x),max z∈Ef(z)g(−z)}=f ⋆ g(x) max z∈Ef(z)g(−z), (iv) for every θ∈Sn−1and for every z, w ∈θ⊥,r, s ∈R, max s∈Rfz(s)gw(r−s) = max s∈Rmin{fz(s)kgwk, gw(r−s)kfzk∞}. As in the previous case, we have that if g(x) = χK(x) is the characteristic function of a convex body, then the (n−1)-th (θ, t)-convolution bodies of the functions are the (n−1)-th θ-convolution bodies of some convex bodies and, as it was proved in [1], if n≥3 we have equality in (i) if and only if for every t∈(0,1) At2(f)(0) = −Kis a simplex and, consequently f(x) = g(−x) is the characteristic function of a simplex. We will prove that in the equality case necessarily one of the functions is the characteristic function of a convex body. Condition (iii) occurs if and only if f ⋆ g(x) or maxz∈Ef(z)g(−z) equals 0 or 1, for every x∈Rn, E ∈An,1. As we have seen in the previous case, if f ⋆ g(x) = 1 for every x∈supp f+ supp gthen fand gare characteristic functions. Let us now assume that there exists x∈supp f+ supp gsuch that f ⋆ g(x)<1. Then for every E∈An,1maxz∈Ef(z)g(−z) equals 0 or 1. Consequently, for every t∈(0,1] At(0) = A1(0) because otherwise there exists some t∈(0,1) and some z∈Rnsuch that t≤f(z)g(−z)<1 and since z /∈ A1(0) there exist 1-dimensional affine subspaces passing through zand not intersecting A1(0) and for all such subspaces we would have t≤maxz∈Ef(z)g(−z)<1. Like before, in such case (i) also holds for t= 1. Notice that if W1(A1(0)) = W1(At(0)) = 0, then for every t∈(0,1) we have that for every θ∈(0,1) Cθ,t ={x∈suppf +suppg :At(x)6=∅}, ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 19 contradicting (i). Thus, if we have equality in (8), W1(A1(0)) >0. Now, if (i) holds, we fix t∈(0,1] and take x∈supp(f) + supp(g). If x /∈ C0,t then At(x) = ∅. If x∈ C0,t then there exists θx∈[0,1] such that x∈∂Cθx,t and x= (1 −θ 1 n x)yfor some y∈∂C0,t. Thus, we have equality in θ 1 n xW1(At(0)) 1 n−1=W1(At(x)) 1 n−1=W1(At(θ 1 n x0 + (1 −θ 1 n x)y)) 1 n−1 ≥θ 1 n xW1(At(0)) 1 n−1+ (1 −θ 1 n x)W1(At(y)) 1 n−1 ≥θ 1 n xW1(At(0)) 1 n−1, and, by the equality cases in Brunn-Minkowski inequality for quermaßintegrtals, if n≥3 then At(x) is homothetic to At(0). Thus, for every x∈Rnand t∈(0,1], At(x) is either empty or a homothetic copy of At(0). Particularizing at t= 1, we have that for any θ∈[0,1] Cn−1 θ,1is the (n−1)-th θ-convolution of the convex bodies Cn−1 θ,1=A1(f)(0) +n−1,θ A1(g)(0). Thus, if (i) holds then by the characterization of the equality cases in [1] A1(f)(0) and −A1(g)(0) are the same simplex A1(0). Now, if we assume that neither of the functions f, g is a characteristic function, with the same proof as before we find a contradiction. Thus, for gmust be a characteristic function and, consequently, f(x) = g(−x) is the characteristic function of a simplex.  6. Colesanti’s inequality for two functions In this section we show that Colesanti’s functional version of Rogers-Shephard inequality (7) can be extended to the case in which we consider any pair of functions and not necessarily g(x) = ¯ f(x). Let us recall that similar results were obtained in [3]. Proof of Theorem 2.4. For any z∈Rnsuch that f⊕g(z)>0 let xz, yz∈Rnbe such that f⊕g(z) = pf(xz)g(yz) with 2z=xz+yz. Notice that xzand yzexist by Remark 2, since f⊕g(z) = pf ⋆ g(2z). Using the log-concavity of fand g, (12) f(x)g(z−x)≥pf(xz)g(yz)pf(2x−xz)g(xz−2x) for every x∈Rn. Integrating in x∈Rn f∗g(z)≥pf(xz)g(yz)ZRnpf(2x−xz)¯g(2x−xz)dx =1 2nf⊕g(z)ZRnpf(x)¯g(x)dx. Integrating in z∈Rnwe finally obtain ZRn f(x)dx ZRn g(x)dx ≥1 2nZRn f⊕g(z)dz ZRnpf(x)¯g(x)dx, as wanted. Let us now characterize the equality cases. If (i) and (ii) are satisfied, then there exists p∈Rnsuch that supp f= supp ¯g=p+C 20 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA and for every z∈supp f+ supp gwe have that supp f∩(z−supp g) = (p+C)∩(z+p+C) = p+C∩(z+C) and since Chas a simplicial section, this equals supp f∩(z−supp g) = p′+C for some p′∈Rn. We will show that p′=p 2+xz 2for xzsuch that f⊕g(z) = pf(xz)g(2z−xz). Notice that, as before, such xzexists, because of the continuity properties of fand gon their supports. In such case we would have that for every z∈supp f+ supp g supp f∩(z−supp g) = xz 2+1 2(supp f∩supp ¯g) and then for every z∈supp f+ supp gand every x∈supp f∩(z−supp g) •2x−xz∈supp f, and •xz−2x∈supp g. Then, inequality (12) holds with equality for the functions f(x) = c1e−ha,xi,x∈ supp fand g(x) = c2e−hb,xi,x∈supp gfor every x, z ∈Rn. In order to show that for every z∈supp f+ supp gwe have p′=p 2+xz 2, notice that 2(supp f+ supp g) = 2C−2C=C−C= supp f+ supp g and so, 2z∈supp f+ supp gand for every x∈supp f∩(2z−supp g) pf(x)g(2z−x) = √c1c2e−ha, x 2ie−hb,z−x 2i =√c1c2e−hb,zie−ha−b, x 2i and so, f⊕g(z) = √c1c2e−hb,zie−min{ha−b, x 2i:x∈supp f∩(2z−supp g)} =√c1c2e−hb,zie−min{ha−b,¯xi:¯x∈p 2+C∩(z+C)}. Since (C∩z+C) = −p+ supp f∩(z−supp g) = −p+p′+Cwe have that min ¯x∈p 2+(C∩z+C)ha−b, ¯xi= min ¯x∈− p 2+p′+Cha−b, ¯xi and since ha−b, xi ≥ 0 for every x∈C, the minimum is attained when ¯x=xz 2= −p 2+p′. Thus p′=p 2+xz 2. Let us now prove that (i) and (ii) are necessary conditions for equality in (11) to hold. We can assume, without loss of generality, that f(x0) = kfk∞=kgk∞= g(y0) = 1 and let us write f(x) = e−u(x)and g(x) = e−v(x)for some convex functions u, v. Notice that, since kfk∞=kgk∞= 1, uand vtake values in [0,+∞]. Equality in (11) happens if and only if for every z∈supp f+supp gand every x∈ supp f∩(z−supp g) we have equality in (12). Thus, for every z∈supp f+supp g the support of both functions as functions of x∈Rnmust be the same and so supp f∩(z−supp g) = xz 2+1 2(supp f∩(−supp g)), where xzis such that f⊕g(z) = pf(xz)g(2z−xz). Notice that in particular this implies that supp f∩(−supp g) is full-dimensional, since we are assuming that supp fand supp gare full-dimensional and then there exists some z∈supp f+ ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 21 supp gsuch that supp f∩(z−supp g) is full-dimensional and then (supp f∩ (−supp g)) is full-dimensional. Besides, for every z∈supp f+ supp gwe have equality in (12) for every x∈ supp f∩(z−supp g) if and only if u1 2(2x−xz) + xz 2=1 2(u(2x−xz) + u(xz)) v1 2(xz−2x) + yz 2=1 2(v(xz−2x) + v(yz)) for every x∈supp f∩(z−supp g), where xz, yz∈Rnare such that f⊕g(z) = pf(xz)g(yz). In particular, for z0=x0+y0 2we have that f⊕g(z0) = pf(x0)g(y0) and so u1 2(2x−x0) + x0 2=1 2u(2x−x0) v1 2(x0−2x) + y0 2=1 2v(x0−2x). Consequently, for every x′=x−x0and y′=x0 2−y0 2−x u(x0+x′) = 1 2u(x0+ 2x′) v(y0+y′) = 1 2v(y0+ 2y′). Thus, supp f∩(z0−supp g) = x0 2+1 2(supp f∩(−supp g)) is a closed convex cone x0+C(with Ca cone with vertex at 0) and so supp f∩(−supp g) = x0+C and for every z∈supp f+ supp g supp f∩(z−supp g) = xz 2+x0 2+C. Furthermore, supp g∩(z0−supp f) = z0−supp f∩(z0−supp g) = z0−x0−C is a closed convex cone y0+C′. This implies that C′=−C. Besides, if the vertex of the cone is unique, z0−x0=y0and then y0=−x0which implies z0= 0. If the vertex is not unique then, since both z0−x0and y0are vertices of the cone y0−C, also y0+ 2(z0−x0−y0) = y0−2z0=−x0and so y0−C=−x0−Cand −x0+C=y0+C. On the other hand, for any z∈supp f+ supp gif x∈supp f∩(z−supp g), then 2x−xz∈x0+C. If equality holds in (12) then uis affine in any segment that connects x0+Cwith a point xzand vis affine in any segment that connects −x0−Cand some yz. Let us now take z∈x0+ supp g. Then −x0+z∈supp gand, since x0∈supp f we have that −x0+z∈(z−supp f)∩supp g=z−supp f∩(z−supp g) = z−xz 2−x0 2−Cand so xz 2∈x0 2−Cand xz∈x0−C. Consequently xz=x0. Otherwise, consider the ray from xzthat passes through x0. Since xz∈x0−Cany point pin this ray such that the segment [xz, p] contains x0 22 DAVID ALONSO, BERNARDO GONZ´ ALEZ, C. HUGO JIM´ ENEZ, RAFAEL VILLA is contained in x0+Cand since u≥0 is affine in the segment [xz, p] and u(x0) = 0 then u= 0 for any such p, contradicting the integrability of f. Thus, for any z∈x0+ supp gwe have that xz=x0and yz= 2z−x0. Notice that then for every z∈x0+ supp gwe have that supp f∩(z−supp g) = xz 2+x0 2+C=x0+C as xz=x0. Consequently, supp f= supp f∩(−supp g) since for every y∈supp f, as Cis a full-dimensional cone, we can take w∈Csuch that y∈ −w+ (x0+C). Thus, if we take z=−w∈ −C=x0−(x0+C)⊂x0+ supp g, we have that y∈z+(x0+C)⊂z−supp gand so y∈supp f∩(z−supp g) = supp f∩(−supp g). Consequently supp f=x0+C⊆ −supp g. Analogously, take z∈ −x0+ supp f. Since −x0∈supp g, we have that x0+z∈ supp f∩(z−supp g) = xz 2+x0 2+Cand, consequently yz 2∈ −x0 2+C. Thus yz∈ −x0+C=y0+C. Consequently yz=y0. Otherwise, consider the ray from yzthat passes through y0. Since yz∈y0+Cany point pin this ray such that the segment [yz, p] contains y0is contained in y0−Cand since v≥0 is affine in the segment [yz, p] and v(x0) = 0 then v= 0 for any such p, contradicting the integrability of g. Thus, for any z∈ −x0+ supp fwe have that xz= 2z−y0and yz=y0. Notice that then for every z∈ −x0+ supp fwe have that supp f∩(z−supp g) = xz 2+x0 2+C=z−y0 2+x0 2+C=z+x0+C, since x0+C=−y0+C. Consequently, −supp g= supp f∩(−supp g) since for every y∈ −supp g, as Cis a full-dimensional cone, we can take w∈Csuch that y∈ −w+ (x0+C). Thus, if we take z=w∈C=−x0+ supp gwe have that y+z∈supp f∩(z−supp g) = z+x0+Cand so y∈x0+C= supp f. Consequently −supp g⊆supp fand so −supp g= supp f=x0+C. Now let us see that vis affine on −x0−C. Let us take x, y ∈ −x0−Cand consider z=y 2+x0 2∈ −C=x0+ supp g. Then, for this z,yz=yand since x∈ −x0−C,vis affine in the segment that connects xand y. Consequently, vis affine on −x0−Cand so g(x) = c2e−hb,xion −x0−C. Thus, Cdoes not contain any straight line lsince otherwise, as vis affine and positive, it must be constant on the line land so Chas only one vertex and x0=−y0and hb, xi<0 for every x∈C\{0}. Analogously, uis also affine on x0+Csince for any x, y ∈x0+C, if we take z=x 2−x0 2∈C=−x0+ supp fwe have that for this z,xz=x−x0−y0=xand so, uis affine on x0+Cand f(x) = c1e−ha,xion x0+C. Since kfk∞=f(x0) we have that ha, xi>0 for every x∈C. Finally, considering the section of Cby a hyperplane, since the intersection of this section with any of its translates is homothetic to itself, the section must be a simplex.  Remark. Theorem 2.4 becomes inequality (7) if g(x) = ¯ f(x) because f⊕g(z) = ∆f(z). Moreover, it also recovers (4) when we particularize f(x) = e−hK(x), g(x) = e−hL(−x), where hKand hLare the support functions of two convex bodies Kand Lthat contain the origin. Then f⊕g(x) = e−hK∩L(x)and pf(x)g(x) = e−hK−L 2 (x), ROGERS-SHEPHARD INEQUALITY FOR LOG-CONCAVE FUNCTIONS 23 and since ZRn e−hK(x)dx =n!|K◦|, then (11) becomes |(K∩L)◦|K−L 2◦≤2n|K◦||L◦|. From the characterization of the equality cases in (11), this only converges to equality when we consider sequences of sets (K◦ n)nand (−L◦ n)nconverging to simplices with 0 in one of the vertices and the same outer normal vectors at the facets that pass through the origin. Taking into account that (K∩L)◦= conv{K◦, L◦}and K−L 2⊂conv{K, −L}, if we change the role of K◦and L◦by Kand −Lfor simplicity, then |K∩L||conv{K, −L}| ≤ K◦−L◦ 2◦|conv{K, −L}| ≤ 2n|K||L|, showing the assertion and slightly strengthening (4). In order to have equality in (4) we must have Kand Lbe simplices with 0 in one of the vertices and the same outer normal vectors at the facets that pass through the origin and K+L 2= conv{K, L}. Thus K=Lis a simplex. Otherwise there exists a direction θ∈Sn−1such that hK(θ)< hL(θ) (or hL(θ)< hK(θ)). Thus, hK+L 2(θ)< hL(θ)≤hconv{K,L}(θ) (or hK+L 2(θ)< hK(θ)≤hconv{K,L}(θ)). ACKNOWLEDGEMENTS Part of this work was carried out at the ‘Instituto de Matem´aticas de la Universidad de Sevilla’ (IMUS) where the D. Alonso was invited and B. Gonz´alez fulfilled the program ‘Ayudas para estancias cortas postdoctorales en el IMUS’, and they are thankful for the invitation and for the good working conditions and environment there. D. Alonso is partially supported by ‘Institut Universitari de Matem`atiques i Aplicacions de Castell´o’, Spanish Ministry of Sciences and Innovation (MICINN) project MTM2013-42105-P and BANCAJA project P1-1B2014-35. B. Gonz´alez is partially supported by Spanish Ministry of Economy and Competitiveness (MINECO) project MTM2012-34037. C. H. Jim´enez and R. Villa are supported by MINECO project MTM2012-30748 and C. H. Jim´enez is also supported by Capes and IMPA. References [1] Alonso-Guti´ errez D., Gonz´ alez Merino B., Jim´ enez C.H. Volume inequalities for the i-th convolution bodies. J. Math. Anal. Appl. 424 (1) (2015), pp. 385-401. [2] Alonso-Guti´ errez D., Jim´ enez C.H., Villa R. Brunn-Minkowski and Zhang inequalities for convolution bodies. Adv. in Math. 238 (2013), pp. 50–69. [3] Arstein S., Einhorn K., Florentin D.I., Ostrover Y. On Godbersen’s conjecture. Geom. Dedicata 178 (1) (2015), pp. 337-350. [4] Arstein S., Giannopoulos A., Milman V. 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