Brunn-Minkowski and Zhang inequalities for convolution bodies

arXiv:1112.4757v2 [math.FA] 11 Feb 2013 BRUNN-MINKOWSKI AND ZHANG INEQUALITIES FOR CONVOLUTION BODIES DAVID ALONSO-GUTI´ ERREZ∗, C. HUGO JIM´ ENEZ†‡, AND RAFAEL VILLA† Abstract. A quantitative version of Minkowski sum, extending the definition of θ-convolution of convex bodies, is studied to obtain extensions of the Brunn-Minkowski and Zhang inequalities, as well as, other interesting properties on Convex Geometry involving convolution bodies or polar projection bodies. The extension of this new version to more than two sets is also given. Zhang inequality, Brunn-Minkowski inequality, Convolution Body. 1. Introduction and motivation The Minkowski sum of two sets A, B ⊆Rnis defined as the set A+B={x∈Rn:A∩(x−B)6=∅}. The essential sum in terms of measure is defined as A+eB={x∈Rn:|A∩(x−B)|>0}, for A, B ⊆Rnmeasurable sets, when | · | denotes the Lebesgue measure in Rn. A quantitative version of this definition, involving the proportional measure of the intersections, gives the following subset of A+B A+θB={x∈A+B:|A∩(x−B)| ≥ θM(A, B)} for θ∈[0,1], whenever M(A, B) := sup x∈A+B |A∩(x−B)|is finite. This set is called the θ-convolution set of Aand B. Note that A+0Bis the usual Minkowski sum A+B. This set is studied for symmetric convex bodies1in [1, 2,3,4] and [5], where the term convolution body is first introduced. However, our notation differs from the one used there, in order to emphasize the connection with the standard Minkowski sum. Properties of θ-convolution bodies are given in Section 2. †Authors partially supported by the Spanish Ministerio de Econom´ıa y Competitividad grant MTM2012-30748 and by the Junta de Andaluc´ıa, grant P08-FQM-03543. ‡Author partially supported by CONACyT. ∗Spanish grant MTM2010-16679 and ”Programa de Ayudas a Grupos de Excelencia de la regi´on de Murcia”, Fundaci´on S´eneca, 04540/GERM/06. 1A convex body is a compact convex subset of Rnwith non-empty interior. 1 2 B-M AND ZHANG INEQUALITIES FOR CONVOLUTION BODIES Recently, there is an increasing interest in finding extensions of the classical integral geometry of the motion group in Euclidean spaces to the group of translations (see [6] and the references therein), motivated by possible applications to the stochastic geometry of homogeneous random geometric structures. Our purpose is to find volume estimates, from above and below, of the θconvolution of two sets. In what follows we will motivate our interest in studying the volume of this family of sets. The celebrated Brunn-Minkowski inequality states that |K+L|1 n≥ |K|1 n+|L|1 n for two convex bodies K, L ⊂Rn, with equality if and only if Kand Lare homothetic. Recall that Kand Lare called homothetic if L=z+λK for some z∈Rnand λ > 0. Two classical references in this topic are [7] and [8]. Brunn-Minkowski inequility has been widely applied to solve a large number of problems involving geometrical quantities such as volume, surface area, and mean width. In the last thirty years the Brunn-Minkowski inequality has become an essential analytical tool to develop the so-called Local Theory of Normed Spaces and Convex Geometric Analysis [9,8,7,10]. Extensions of this inequality for non convex sets, even for non-measurable sets, have been also studied. We refer only to [11] for details and references. In Section 3a generalization of the Brunn-Minkowski inequality is studied. Even though extensive work with this inequality as backbone has emerged both within the class of convex bodies [12,13,14] and under other settings [15,16,17,18], we pursuit something closer in spirit to [19,20]. See [21] for a comprehensive survey on the Brunn-Minkowski inequality including extensions, applications and its relation to other analytical inequalities. Namely, we pose the problem of finding the best function ϕn(θ) such that (1.1) |K+θL|1 n≥ϕn(θ)1 n(|K|1 n+|L|1 n) for any convex bodies K, L ⊂Rn. It is proved that ϕn(θ) = (1 −θ1 n)n satisfies (1.1). Some particular cases are also studied. Following the work of Kiener [1], Schmuckensl¨ager [2] proved that for any convex body Kof volume 1, (1.2) (1 −θ)Π∗(K)⊆K+θ(−K)⊆log 1 θΠ∗(K) where Π∗(K) is the polar projection body of K, the unit ball of the norm kxkΠ∗(K)=|x||Px⊥K|. Here Px⊥denotes the orthogonal projection on the hyperplane orthogonal to x. These inclusions imply |K|Π∗(K) = lim θ→1− K+θ(−K) 1−θin the Haussdorff metric for any convex body K. B-M AND ZHANG INEQUALITIES FOR CONVOLUTION BODIES 3 In Section 4we modify the argument to improve the estimate (1.2) (see Proposition 4.1) (1 −θ)|K|Π∗(K)⊆K+θ(−K)⊆n(1 −θ1 n)|K|Π∗(K). The most famous inequality concerning the volume of the polar projection body of a convex body K⊂Rnis Petty projection inequality, |K|n−1|Π∗(K)| ≤ ωn ωn−1n where ωndenotes the volume of the n-dimensional Euclidean ball. The equality is attained provided Kis an ellipsoid (See [22]). A different proof using convolutions can be found in [4]. In [23], Zhang proved a reverse form of this inequality (1.3) |K|n−1|Π∗(K)| ≥ 1 nn2n n for any convex body, with equality if and only if Kis a simplex. Zhang inequality can be written as (1.4)  lim θ→1− K+θ(−K) 1−θ ≥1 nn2n n|K|. It is worth mentioning that Tsolomitis studies in [5] the behavior of limiting convolution bodies (1.5) lim θ→1− K+θL (1 −θ)α for symmetric convex bodies Kand L, and some exponent α, giving some regularity conditions under which (for some specific α) the limit (1.5) is non-degenerated, denoted by C(K, L). In [24], Rogers and Shephard obtained the inequality (1.6) |K−K| ≤ 2n n|K| for any convex body K, with equality if and only if Kis a simplex. Throughout the proof it is showed that (1.7) K+θ(−K)⊇(1 −θ1 n)(K−K) with equality if and only if Kis a simplex. They also showed in [25] the extension for two different convex bodies (1.8) |K−L||K∩L| ≤ 2n n|K||L|. The last part of Section 4is devoted to generalize the inclusions stated in (1.2) and Zhang inequality (1.3) for limiting convolutions of different convex bodies. This generalization is a consequence of Corollary 2.5, from which (1.8) can be obtained (see Proposition 4.7). 4 B-M AND ZHANG INEQUALITIES FOR CONVOLUTION BODIES Given that some classical geometric inequalities are recovered and extended through the use of convolution bodies, it would be natural to consider the extension of the convolution body of more than two bodies in order to generalize these inequalities. In Section 5we study such an extension obtaining similar inequalities when we consider more than two bodies. Surprisingly, it turns out when studying the equality cases, that these inequalities can only be sharp when convoluting two convex bodies and not when considering three or more bodies. 2. Properties of the θ-convolution of convex bodies In this section we give some properties of the θ-convolution of two convex bodies, from which the Brunn-Minkowski-type inequality for the θconvolution of convex bodies |K+θL|1 n≥(1−θ1 n)|K|1 n+|L|1 nwill follow. However, this bound is not sharp, as we will see below. We now list some basic properties of M(K, L) and the θ-convolution in the following Proposition 2.1. Let K, L ⊂Rnbe compact sets, λ≥0,x∈Rnand T∈GLn(R) = {T:Rn→Rn:Tis linear}. (a1) M(K, L) = M(L, K). (a2) M(x+K, L) = M(K, L). (a3) M(λK, λL) = λnM(K, L). (a4) M(TK, T L) = |det T|M(K, L). (a5) If K=−Lor K, L are symmetric, then M(K, L) = |K∩(−L)|. Let θ∈[0,1]. (b1) (λK) +θ(λL) = λ(K+θL). (b2) K+θL=L+θK. (b3) (x+K) +θL=x+ (K+θL). (b4) TK +θTL =T(K+θL). A first question about this θ-convolution is its convexity, provided that K and Lare both convex. The affirmative answer is a consequence of the following result. In what follows, using (b3) in Proposition 2.1 above, we will assume without loss of generality, that (2.1) M(K, L) = |K∩(−L)|. Proposition 2.2. Let K, L ⊂Rnbe convex bodies satisfying (2.1). Then for every θ1, θ2, λ1, λ2∈[0,1] such that λ1+λ2≤1we have that (2.2) λ1(K+θ1L) + λ2(K+θ2L)⊆K+θL, where 1−θ1 n=λ1(1 −θ 1 n 1) + λ2(1 −θ 1 n 2). B-M AND ZHANG INEQUALITIES FOR CONVOLUTION BODIES 5 Proof. Let x1∈K+θ1Land x2∈K+θ2L. From the general inclusion K∩(λ0A0+λ1A1+λ2A2)⊃λ0K∩A0+λ1K∩A1+λ2K∩A2 where Kis convex and λ0+λ1+λ2= 1, and using the convexity of L, we have (2.3) K∩(λ1x1+λ2x2−L)⊇(1−λ1−λ2)[K∩(−L)]+λ1[K∩(x1−L)]+λ2[K∩(x2−L)]. Taking volumes, using the classical Brunn-Minkowski inequality and the fact that xi∈K+θiLwe have (2.4) |K∩(λ1x1+λ2x2−L)| ≥ [1 −λ1(1 −θ 1 n 1)−λ2(1 −θ 1 n 2)]nM(K, L), which proves that λ1x1+λ2x2∈K+θLfor θ= [1 −λ1(1 −θ 1 n 1)−λ2(1 − θ 1 n 2)]n. Taking θ1=θ2and λ2= 1 −λ1we have Corollary 2.3. Let K, L ⊂Rnbe convex bodies and θ∈[0,1]. Then K+θL is convex. The following result on K+1Lwill be used later on, and it is a consequence of Proposition 2.2. Corollary 2.4. Let K, L ⊂Rnbe convex bodies. For any x∈K+1L, K∩(x−L)is a translation of K∩(−L). Proof. Taking θ1=θ2= 1 in Proposition 2.2, we get θ= 1, and then Brunn-Minkowski inequality (2.4), obtained from (2.3), holds with equality. Then (see equality cases in Brunn-Minkowski inequality in [26]) the four sets involved in (2.3), K∩(λ1x1+λ2x2−L), K ∩(−L), K ∩(x1−L), K ∩(x2−L), are all homothetic. As they all have the same volume (equals to M(K, L)), homotheties are indeed translations.  Taking θ1=θ2leads us to the following Corollary 2.5. Let K, L ⊂Rnbe convex bodies satisfying (2.1). Then, for every 0≤θ0≤θ < 1we have K+θ0L 1−θ0 1 n ⊆K+θL 1−θ1 n . Proof. Taking θ1=θ2=θ0in the above proposition, for any λ1, λ2∈[0,1] such that λ1+λ2≤1 λ1(K+θ0L) + λ2(K+θ0L)⊆(λ1+λ2)(K+θ0L)⊆K+θL, 6 B-M AND ZHANG INEQUALITIES FOR CONVOLUTION BODIES with 1 −θ1 n= (λ1+λ2)(1 −θ0 1 n). Since λ1+λ2=1−θ1 n 1−θ 1 n 0 , 1−θ1 n 1−θ 1 n 0 (K+θ0L)⊆K+θL whenever λ1+λ2≤1, which means 0 ≤θ0≤θ≤1.  The following extension of (1.7) from Rogers and Shephard’s work will be used to get a first Brunn-Minkowski-type inequality. Corollary 2.6. Let K, L ⊂Rnbe convex bodies. Then (2.5) θ1 n(K+1L) + (1 −θ1 n)(K+L)⊆K+θL. Proof. Take θ1= 1, θ2= 0, λ1=θ1 n,and λ2= (1 −θ1 n) in Proposition 2.2 to obtain the desired result.  Note that the condition 0 ∈K+1Lis equivalent to M(K, L) = |K∩(−L)|, and that is verified under the assumptions of Proposition 2.1 (a5). We will use the following description of the boundary of K+θL. Lemma 2.7. Let K, L ⊂Rnbe convex bodies, and θ∈[0,1). Then ∂(K+θL) = {x∈K+L:|K∩(x−L)|=θM(K, L)}. In particular, for θ= 0, ∂(K+L) = {x∈K+L:|K∩(x−L)|= 0}. Consequently, for any x∈K+L,x6∈ K+1L, there exists a unique θ∈[0,1) such that x∈∂(K+θL). Proof. Let f:Rn→[0,+∞) be given by f(x) = |K∩(x−L)|1 n. Using properties of the Lebesgue measure and Brunn-Minkowski inequality, it can be deduced that fis continuous on Rnand concave on K+L. It is then clear that K+θLis closed. The first assertion is equivalent to int(K+θL) = {x∈K+L:f(x)n> θM(K, L)}. Since the right-hand set is open and it is contained in K+θL, it remains to be shown the inclusion int(K+θL)⊆ {x∈K+L:f(x)n> θM(K, L)}. Let us take any x∈K+L, with f(x)n=θM(K, L). It is left to show that x6∈ int(K+θL). We will assume without loss of generality that (2.1) is satisfied. For any λ∈(0,1), write x=λ(λ−1x) + (1 −λ)·0. If λ−1x∈K+L, using the concavity of fon K+L, (θM(K, L)) 1 n=f(x)≥λf(λ−1x)+(1−λ−1)f(0) = λf(λ−1x)+(1−λ−1)M(K, L)1 n B-M AND ZHANG INEQUALITIES FOR CONVOLUTION BODIES 7 and therefore f(λ−1x)n≤θ1 n−(1 −λ−1) λM(K, L)1 n<(θM(K, L)) 1 n. Then λ−1x6∈ K+θLfor any λ < 1 (for λ−1x /∈K+Lit is trivial). Hence x6∈ int(K+θL). To obtain the last assertion, just take θ=|K∩(x−L)| M(K, L)∈[0,1).  Equality cases in Proposition 2.2 and Corollary 2.6 are stated in the following result. Proposition 2.8. Let K, L ⊂Rnbe convex bodies. The following conditions are equivalent. (i) For every θ1, θ2, λ1, λ2∈[0,1] such that λ1+λ2= 1, we have λ1(K+θ1L) + λ2(K+θ2L) = K+θL, where θ1 n=λ1θ 1 n 1+λ2θ 1 n 2. (ii) For every θ∈[0,1],θ1 n(K+1L) + (1 −θ1 n)(K+L) = K+θL. (iii) Kand −Lare homothetic n-simplices. For the proof, we will use the following result from [27] (we state it here in the form it will be used in this paper). Theorem 2.9. (Soltan) Let K, L ⊂Rnbe convex bodies. The following conditions are equivalent: (i) Kand Lare homothetic n-simplices. (ii) The n-dimensional intersections K∩(z−L),z∈Rn, are all homothetic to K∩(−L). Proof of Proposition 2.8.Using a translation we may assume that (2.1) is satisfied. Conditions (i) and (ii) are equivalent. Indeed, (i) trivially implies (ii). Suppose (ii) holds. Then, using the equality for θ1,θ2and θsuccessively, and the convexity of the convolution sets, λ1(K+θ1L) + λ2(K+θ2L) =λ1θ1 1 n(K+1L) + (1 −θ1 1 n)(K+L) +λ2θ2 1 n(K+1L) + (1 −θ2 1 n)(K+L) = (λ1θ 1 n 1+λ2θ 1 n 2)(K+1L) + (λ1(1 −θ 1 n 1) + λ2(1 −θ 1 n 2))(K+L) =K+θL, and (i) follows. 8 B-M AND ZHANG INEQUALITIES FOR CONVOLUTION BODIES Suppose (ii) holds, and take x∈K+L,x6∈ ∂(K+L). If x∈K+1L, Corollary 2.4 shows that K∩(x−L) is a translation of (so homothetic to) K∩(−L). If x6∈ K+1L, by Lemma 2.7, there is a θ∈(0,1) such that x∈∂(K+θL). Using (ii), x=θ1 nx1+ (1 −θ1 n)x2, for some x1∈K+1L,x2∈K+L. Now, inclusion (2.3) and inequality (2.4) are both equalities (since |K∩(x− L)|=θM(K, L)), and then K∩(x−L), K∩(x1−L) and K∩(x2−L) are all homothetic (see equality cases in Brunn-Minkowski inequality in [26]). Since x1∈K+1L, they are all homothetic to K∩(−L). Then, all the n-dimensional intersections K∩(x−L) are homothetic to the same body K∩(−L). Now it follows from Theorem 2.9 that K,−Land K∩(−L) are homothetic simplices. Now suppose condition (iii) holds. After an affine transformation we may assume that K=(t∈Rn:ti≥0, n X i=1 ti≤1) and −L=λK with 0 < λ ≤1. For any x∈Rn, K∩(x−L) = (t∈Rn:tj≥x+ j, n X i=1 ti≤min{1, λ + n X i=1 xi}), where r+= max{r, 0}. Then, K∩(x−L) = z(x)+λ(x)K, where zj(x) = x+ j, and λ(x) = min{1, λ + n X i=1 xi} − n X i=1 x+ i whenever λ(x)≥0. For those x∈Rn, we have |K∩(x−L)|=λ(x)n|K|. It is easy to see that λ(x)≤λ, and equality holds if and only if xj≥0 for all jand λ+Pn i=1 xi≤1. Then M(K, L) = λn|K|, and K+1L= (1 −λ)K. Using absolute values, λ(x) can be rewritten as λ(x) = 1 2 1 + λ− 1−λ− n X i=1 xi − n X i=1 |xi|!. Then K+θL=(x∈Rn: 1−λ− n X i=1 xi + n X i=1 |xi| ≤ 1 + λ(1 −2θ1 n)). In particular, letting θ= 0, we obtain an expression for K+L. In order to prove (ii), it is enough to prove the inclusion K+θL⊆θ1 n(K+1L) + (1 −θ1 n)(K+L) for every θ∈(0,1). Any x∈K+θLsatisfies (2.6)  1−λ− n X i=1 xi + n X i=1 |xi| ≤ θ1 n(1 −λ) + (1 −θ1 n)(1 + λ). B-M AND ZHANG INEQUALITIES FOR CONVOLUTION BODIES 9 Recall that  1−λ− n X i=1 ai + n X i=1 |ai|= 1 −λfor any a∈K+1L. Consequently, if (2.7) x=θ1 na+ (1 −θ1 n)b, with a∈K+1L, then b∈K+Lprovided that the left hand side in (2.6)  θ1 n(1 −λ− n X i=1 ai) + (1 −θ1 n)(1 −λ− n X i=1 bi) + n X i=1 |θ1 nai+ (1 −θ1 n)bi| equals θ1 n 1−λ− n X i=1 ai + (1 −θ1 n) 1−λ− n X i=1 bi +θ1 n n X i=1 |ai|+ (1 −θ1 n) n X i=1 |bi|. Considering equality cases in triangle inequality, this happens provided that 1−λ−Pn i=1 aihas the same sign as 1 −λ−Pn i=1 bi, and for any j,ajhas the same sign as bj(here r, s have the same sign iff r·s≥0). If Pn i=1 x+ i≤1−λ, it is enough to consider a∈K+1Lin (2.7), so that aj=x+ j. Then bj= (1 −θ1 n)−1xjif xj<0 and bj=xjif xj≥0. In any case, bjhas the same sign as aj, and therefore b∈K+L. If Pn i=1 x+ i>1−λ, it is enough to take a∈K+1Lin (2.7), so that aj= (1 −λ)x+ j Pn i=1 x+ i . Then b∈K+Lprovided that bj≥0 for those jso that xj>0. But in that case xj=θ1 n(1 −λ) Pn i=1 x+ i xj+ (1 −θ1 n)bj Then bj≥0 provided that θ1 n(1 −λ) Pn i=1 x+ i ≤1, which is assumed to be true. In any case, we can find a decomposition as (2.7), which proves (ii).  Now we can deduce equality cases in Corollary 2.5. Proposition 2.10. Let K, L ⊂Rnbe convex bodies satisfying (2.1). The following conditions are equivalent. (i) For any 0≤θ0≤θ≤1,(1 −θ1 n)(K+θ0L) = (1 −θ 1 n 0)(K+θL). (ii) For every θ∈[0,1],(1 −θ1 n)(K+L) = K+θL. (iii) K=−Lis an n-simplex. Proof. Again, conditions (i) and (ii) are easily seen to be equivalent. 16 B-M AND ZHANG INEQUALITIES FOR CONVOLUTION BODIES If equality holds, then (ii) in Proposition 2.10 is satisfied, and therefore K=−Lis a simplex.  5. Convolution of mbodies In this section we will extend the definition of θ-convolution bodies to more than two sets. The θ-convolution is not associative (as a simple computation with Euclidean balls of different radius shows) so a definition of an m-fold convolution can not be made inductively. Nevertheless, since |K∩(x−L)|= χK∗χL(x) and the convolution is associative, it seems natural to make the following extension of θ-convolution bodies: Definition 5.1. Let K1,...,Kmbe mmeasurable sets in Rnand let θ∈ [0,1]. We define their θ-convolution as the set K1+θ···+θKm={x∈K1+···+Km:χK1∗· · ·∗χKm(x)≥θM(K1,...,Km)} when M(K1,...,Km) = maxx∈RnχK1∗ · · · ∗ χKm(x) is finite. For θ= 0 the set K1+0··· +0Kmis just the support of the function χK1∗ · · · ∗ χKm, the usual Minkowski sum K1+···+Km. The commutative and associative properties of the convolution imply trivially that (a1) χK1∗ · · · ∗ χKm=χKσ(1) ∗ · · · ∗ χKσ(m)for any σpermutation of {1, ..., m}. (a2) χz+K1∗ · · · ∗ χKm(x) = χK1∗ · · · ∗ χKm(x−z) (a3) χT K1∗ · · · ∗ χT Km(x) = |det T|m−1χK1∗ · · · ∗ χKm(T−1x) for any T∈GLn(R). Consequently we have the following result, analogous to Proposition 2.1. Proposition 5.2. Let K1,...,Kmbe compact sets in Rn,λ∈R, θ ∈[0,1], x∈Rnand T∈GLn(R). Then: (b1) (λK1) +θ···+θ(λKm) = λ(K1+θ···+θKm) (b2) K1+θ···+θKm=Kσ(1) +θ···+θKσ(m)for any σpermutation of {1, ..., m}. (b3) (x+K1) +θK2+θ···+θKm=x+ (K1+θ···+θKm) (b4) TK1+θ···+θTKm=T(K1+θ···+θKm) The convexity is transmitted to the θ-convolution of mconvex bodies. Proposition 5.3. Let K1,...,Kmbe convex bodies in Rn. Then K1+θ ···+θKmis a convex body. Proof. The characteristic function of each convex body Kiis log-concave. The convolution of log-concave functions is log-concave, and the level sets of log-concave functions are convex.  B-M AND ZHANG INEQUALITIES FOR CONVOLUTION BODIES 17 Corollary 2.5 is proved by using that χK1∗χK2is 1 n-concave in its support. In order to generalize this result, we have to prove that the convolution of more than two characteristic functions is s-concave for some s. We get this result for s−1= (m−1)nby considering sets in dimension (m−1)n. As in the case m= 2, we may assume, without loss of generality, that (5.1) M(K1,...,Km) = χK1∗ · · · ∗ χKm(0). Proposition 5.4. Let K1,...,Km⊂Rnbe convex bodies satisfying (5.1). Then, for any θ1, θ2, λ1, λ2∈[0,1] such that λ1+λ2≤1we have (5.2) λ1(K1+θ1···+θ1Km) + λ2(K1+θ2···+θ2Km)⊆K1+θ···+θKm, where 1−θ1 (m−1)n=λ1(1 −θ 1 (m−1)n 1) + λ2(1 −θ 1 (m−1)n 2). Proof. First at all, notice that for any x∈Rn, χK1∗ · · · ∗ χKm(x) =ZRn χK1∗ · · · ∗ χKm−1(tm−1)χKm(x−tm−1)dtm−1 =ZRnZRn χK1∗... · · · ∗ χKm−2(tm−2)χKm−1(tm−1−tm−2)χKm(x−tm−1)dtm−2dtm−1 =··· =ZRn ···ZRn χK1(t1)χK2(t2−t1)... . . . χKm−1(tm−1−tm−2)χKm(x−tm−1)dt1. . . dtm−1 =|Ωm−1(x)| where Ωm−1(x) = {(t1,...tm−1)∈R(m−1)n:t1∈K1, t2−t1∈K2, . . . . . . tm−1−tm−2∈Km−1, x −tm−1∈Km}. The convexity of K1,...,Kmgives (5.3) Ωm−1(λ1x1+λ2x2)⊇(1 −λ1−λ2)Ωm−1(0) + λ1Ωm−1(x1) + λ2Ωm−1(x2) for any x1, x2∈Rnand λ1, λ2≥0 such that λ1+λ2≤1. In particular, if x1∈K1+θ1···+θ1Kmand x2∈K1+θ2···+θ2Km, Brunn-Minkowski inequality in R(m−1)nimplies (5.4) |Ωm−1(λ1x1+λ2x2)| ≥ ((1−λ1−λ2)+λ1θ 1 (m−1)n 1+λ2θ 1 (m−1)n 2)(m−1)n|Ωm−1(0)|, which shows that λ1x1+λ2x2∈K1+θ···+θKmwith 1−θ1 (m−1)n=λ1(1 −θ 1 (m−1)n 1) + λ2(1 −θ 1 (m−1)n 2). 18 B-M AND ZHANG INEQUALITIES FOR CONVOLUTION BODIES Remark 5.5. The concavity of the function x∈K1+···+Km7→ χK1∗ · · · ∗ χKm(x)1 (m−1)n allows us to write, for any θ∈[0,1), the boundary of K1+θ···+θKmas ∂(K1+θ···+θKm) = {x∈K1+···+Km:χK1∗· · ·∗χKm(x) = θM(K1,...,Km)}. In particular, for θ= 0, ∂(K1+···+Km) = {x∈K1+···+Km:χK1∗ · · · ∗ χKm(x) = 0}. That implies that for any x∈K1+···+Km,x6∈ K1+1···+1Km, there exists a (unique) θ∈[0,1) such that x∈∂(K1+θ···+θKm). Remark 5.6. Taking θ1=θ2= 1, and following the proof of Proposition 5.4, we get equality in the inclusion (5.3) and in the Brunn-Minkowski inequality (5.4). Consequently, Ωm−1(x) are all homothetic to Ωm−1(0) for any x∈K1+1···+1Km. Using that they all have the same volume, we get that they are translations of Ωm−1(0). In particular, taking θ1=θ2and λ2= 1 −λ1in Proposition 5.4, we get the convexity of K1+θ···+θKm. Also, taking θ1=θ2=θ0, we get the version of Corollary 2.5 to mbodies. Corollary 5.7. Let K1,...,Km⊂Rnbe convex bodies such that (5.1) is satisfied. Then for any 0< θ0≤θ < 1 K1+θ0···+θ0Km 1−θ 1 (m−1)n 0 ⊆K1+θ···+θKm 1−θ1 (m−1)n . We can easily get the Brunn-Minkowski type inequality for mbodies, a generalization of Corollary 3.2. Corollary 5.8. Let K1,...,Km⊂Rnbe convex bodies. Then |K1+θ···+θKm|1 n≥1−θ1 (m−1)n(|K1|1 n+···+|Km|1 n). Finally, we get Zhang and Roger-Shephard type inequalities for mbodies. For K1,...,Km⊆Rn, let C1(K1,...,Km) = lim θ→1 K1+θ···+θKm 1−θ. As in the case m= 2, the existence of the previous limit follows from Corollary 5.7. Corollary 5.9. Let K1,...,Km⊆Rnbe convex bodies. Then |K1+···+Km| ≤ m n n|K1|···|Km| M(K1,...,Km)≤(m−1)nnn|C1(K1,...,Km)| Proof. The proof runs as in Theorem 4.6 and Proposition 4.7 resp., using Corollary 5.7, instead of Corollary 2.5. B-M AND ZHANG INEQUALITIES FOR CONVOLUTION BODIES 19 Regarding the study of equality cases, we will show that equality never occur in (5.2) for all λ1, λ2such that λ1+λ2= 1 unless m= 2 or n= 1. That implies that extensions of Zhang, Roger-Shephard and Brunn-Minkowski type inequalities are not sharp for m > 2 and n > 1. The following result can be proved as in Proposition 2.8. Proposition 5.10. Let K1,...,Km⊂Rnbe convex bodies satisfying (5.1) and such that for any θ1, θ2, λ1, λ2∈[0,1] such that λ1+λ2= 1 we have (5.5) λ1(K1+θ1···+θ1Km) + λ2(K1+θ2···+θ2Km) = K1+θ···+θKm, where θ1 (m−1)n=λ1θ 1 (m−1)n 1+λ2θ 1 (m−1)n 2.Then for every x∈K1+···+Km Ωm−1(x)is homothetic to Ωm−1(0) . Then we will show that this consequence can not occur for m≥3. Proposition 5.11. Let m≥3and K1,...,Km⊂Rnbe convex bodies satisfying (5.1). Then it is not possible that Ωm−1(x)is homothetic to Ωm−1(0) for every x∈K1+···+Km. For the proof, we will use the following fact on sum of simplices, which is of independent interest. Lemma 5.12. Let K, L ⊂Rnbe n-dimensional convex bodies. If K+Lis an n-dimensional simplex, then Kand Lare both n-dimensional homothetic simplices. Proof. Denote by hC(x) = max{hx, yi:y∈C}the support function of a compact set C⊂Rn. Let us write K+L= conv{w0,...wn}. We will show that Lis a simplex. Each wiis an extreme point of K+L. Then wi=ui+viwhere ui∈Kand vi∈Lare extreme points of Kand Lresp. In particular, vi=wi−ui∈L and so conv{w0−u0,...wn−un} ⊆ L. Also hK(x) + hL(x) = hK+L(x) = max 0≤i≤nhx, wii = max 0≤i≤n(hx, uii+hx, wi−uii) ≤max 0≤i≤nhx, uii+ max 0≤i≤nhx, wi−uii ≤hK(x) + max 0≤i≤nhx, wi−uii Thus hL(x)≤max 0≤i≤nhx, wi−uii,so L⊆conv{w0−u0,...wn−un}, and L is a simplex. Now, write K= conv{u0,...,un}and L= conv{v0,...vn}. We will prove that they are homothetic simplices. 20 B-M AND ZHANG INEQUALITIES FOR CONVOLUTION BODIES Let Ai={s∈Sn−1:hK(s) = hx, uii} and Bi={s∈Sn−1:hL(s) = hx, vii}.Kand Lare homothetic if and only if both partitions of Sn−1 Sn−1= n [ i=0 Ai= n [ i=0 Bi are identical. Assume they are not the same, then the partition Sn−1= n [ i,j=0 Ci,j where Ci,j ={s∈Sn−1:hK+L(s) = hx, ui+vji} has more than n+ 1 elements, since hK+L(s) = hK(s) + hL(s) = max 0≤i≤nhx, uii+ max 0≤j≤nhx, vji= max 0≤i,j≤nhx, ui+vji. Then K+Lis not a simplex.  Remark 5.13. An argument similar to that used in the remark before Theorem 3.2.3 in [8] leads to the same claim. However, we have included a direct proof of it for the sake of completeness. Proof of Proposition 5.11.Notice that Ωm−1(x) = {(t1,...tm−1)∈R(m−1)n: (t1,...tm−2)∈Ωm−2(tm−1), tm−1∈(x−Km)∩(K1+···+Km−1)}. and that Ωm−2(tm−1) is non-empty if and only if tm−1∈K1+···Km−1. Then the projection onto the tm−1coordinate is Pm−1(Ωm−1(x)) = (x−Km)∩(K1+···+Km−1). Suppose that Ωm−1(x) is homothetic to Ωm−1(0) for every x∈K1+···+Km. Then their projections are also homothetic. Then (x−Km)∩(K1+···+ Km−1) is homothetic to (−Km)∩(K1+···+Km−1) for any x∈K1+···+ Kmand Soltan’s Theorem 2.9 implies that K1+···Km−1and −Kmare homothetic simplices. On the other hand, the projection onto the tm−2coordinate is Pm−2(Ωm−1(x)) = [ tm−1∈x−Km Pm−2(Ωm−2(tm−1)) =[ tm−1∈x−Km (tm−1−Km−1)∩(K1+···+Km−2) = (x−(Km−1+Km)) ∩(K1+···+Km−2) and so we have that for every x∈K1+···+Km, (x−(Km−1+Km)) ∩ (K1+···+Km−2) is homothetic to (−(Km−1+Km)) ∩(K1+···+Km−2). Soltan’s Theorem 2.9 shows again that K1+···+Km−2and −(Km−1+Km) are homothetic simplices. If Kmand Km−1+Kmare both simplices, by Lemma 5.12,Km−1is also a simplex homothetic to Km. But (K1+···+Km−1) + Kmis a simplex, so B-M AND ZHANG INEQUALITIES FOR CONVOLUTION BODIES 21 Lemma 5.12 shows that K1+···+Km−1and Kmare homothetic. But then Kmis a simplex homothetic to −Km, a contradiction, unless n= 1.  Acknowledgements Part of this work was done while the first two named authors were attending the Thematic Program on Asymptotic Geometric Analysis in Fall 2010 at the Fields Institute in Toronto, where the first named author enjoyed the J. Marsden’s postdoctoral fellowship. We appreciate the hospitality. We are indebted to Prof. A. Giannopoulos for many helpful discussions and for pointing out to us several references at the early stages of this work. We also would like to thank Steven Taschuk for pointing out to us the alternative proof of Lemma 5.12 exposed in Remark 5.13. Finally, the authors thank the referee for suggesting the study of equality cases and other improvements in exposition. References [1] K. Kiener, “Extremalit¨at von Ellipsoiden und die Faltungsungleichung von Sobolev,” Archiv der Mathematik, vol. 46, pp. 162–168, 1986. 10.1007/BF01197494. [2] M. Schmuckenschl¨ager, “The distribution function of the convolution square of a convex symmetric body in Rn,” Israel Journal of Mathematics, vol. 78, pp. 309–334, 1992. 10.1007/BF02808061. [3] M. Meyer, S. Reisner, and M. Schmuckenschl¨ager, “The volume of the intersection of a convex body with its translates,” Mathematika, vol. 40, no. 02, pp. 278–289, 1993. [4] M. Schmuckenschl¨ager, “Petty’s projection inequality and Santalo’s affine isoperimetric inequality,” Geom. Dedicata, vol. 57, no. 3, pp. 285–295, 1995. [5] A. Tsolomitis, “Convolution bodies and their limiting behavior,” Duke Math. J., vol. 87, no. 1, pp. 181–203, 1997. [6] R. Schneider, “Mixed polytopes,” Discrete Comput. Geom., vol. 29, no. 4, pp. 575– 593, 2003. [7] T. Bonnesen and W. Fenchel, Theory of convex bodies. Moscow, ID: BCS Associates, 1987. Translated from the German and edited by L. Boron, C. Christenson and B. Smith. [8] R. Schneider, Convex bodies: the Brunn-Minkowski theory, vol. 44 of Encyclopedia of Mathematics and its Applications. Cambridge: Cambridge University Press, 1993. [9] R. J. Gardner, Geometric tomography, vol. 58 of Encyclopedia of Mathematics and its Applications. Cambridge: Cambridge University Press, second ed., 2006. [10] V. D. Milman and G. Schechtman, Asymptotic theory of finite-dimensional normed spaces, vol. 1200 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, 1986. With an appendix by M. Gromov. [11] H. Hadwiger, Vorlesungen ¨uber Inhalt, Oberfl¨ache und Isoperimetrie. Berlin: Springer-Verlag, 1957. [12] W. Firey, “p-means of convex bodies,” Math. Scand., vol. 10, pp. 17–24, 1962. [13] E. Lutwak, “The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem,” J. Differential Geom., vol. 38, no. 1, pp. 131–150, 1993. [14] E. Lutwak, “The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas,” Adv. Math., vol. 118, no. 2, pp. 244–294, 1996. [15] C. Borell, “The Brunn-Minkowski inequality in Gauss space,” Invent. Math., vol. 30, no. 2, pp. 207–216, 1975. 22 B-M AND ZHANG INEQUALITIES FOR CONVOLUTION BODIES [16] R. Vitale, “The Brunn-Minkowski inequality for random sets,” J. Multivariate Anal., vol. 33, no. 2, pp. 286–293, 1990. [17] B. Uhrin, “Curvilinear extensions of the Brunn-Minkowski-Lusternik inequality,” Adv. Math., vol. 109, no. 2, pp. 288–312, 1994. [18] H. J. Brascamp and E. H. Lieb, “On extensions of the Brunn-Minkowski and Pr´ekopa- Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation,” J. Functional Analysis, vol. 22, no. 4, pp. 366–389, 1976. [19] F. Barthe, “Restricted Pr´ekopa-Leindler inequality,” Pacific J. Math., vol. 189, no. 2, pp. 211–222, 1999. [20] S. Szarek and D. Voiculescu, “Volumes of restricted Minkowski sums and the free analogue of the entropy power inequality,” Comm. Math. Phys., vol. 178, no. 3, pp. 563–570, 1996. [21] R. J. Gardner, “The Brunn-Minkowski inequality,” Bull. Amer. Math. Soc. (N.S.), vol. 39, no. 3, pp. 355–405, 2002. [22] C. M. Petty, “Isoperimetric problems,” in Proceedings of the Conference on Convexity and Combinatorial Geometry (Univ. Oklahoma, Norman, Okla., 1971), pp. 26–41, Dept. Math., Univ. Oklahoma, Norman, Okla., 1971. [23] G. Y. Zhang, “Restricted chord projection and affine inequalities,” Geom. Dedicata, vol. 39, no. 2, pp. 213–222, 1991. [24] C. A. Rogers and G. C. Shephard, “The difference body of a convex body,” Arch. Math. (Basel), vol. 8, pp. 220–233, 1957. [25] C. A. Rogers and G. C. Shephard, “Convex bodies associated with a given convex body,” J. London Math. Soc., vol. 33, pp. 270–281, 1958. [26] H. Brunn, “Referat ¨uber eine Arbeitet: Exacte Grundlagen f¨ur eine Theorie der Ovale,” S.-B. Bayer. Akad. Wiss., pp. 93–111, 1894. [27] V. Soltan, “A characterization of homothetic simplices,” Discrete Comput. Geom., vol. 22, no. 2, pp. 193–200, 1999. C. Hugo Jim´ enez and Rafael Villa, Dept. de An´ alisis Matem´ atico, Facultad de Matem´ aticas, Universidad de Sevilla, PO Box 1160, 41080 Sevilla, Spain E-mail address:carloshu[email protected],[email protected] David Alonso-Guti´ errez, Dept. of Math. and Stats., 632 Central Academic Building, University of Alberta, Edmonton, Ab, Canada T6G 2G1 E-mail address:[email protected]