THE LOG-BRUNN-MINKOWSKI INEQUALITY

Transcription

1 THE LOG-BRUNN-MINKOWSKI INEQUALITY Abstract. For origin-symmetric convex bodies (i.e., the unit balls of finite dimensional Banach saces) it is conjectured that there exist a family of inequalities each of which is stronger than the classical Brunn-Minkowski inequality and a family of inequalities each of which is stronger than the classical Minkowski mixed-volume inequality. It is shown that these two families of inequalities are equivalent in that once either of these inequalities is established, the other must follow as a consequence. All of the conjectured inequalities are established for lane convex bodies. 1. Introduction The fundamental Brunn-Minkowski inequality states that for convex bodies K, L in Euclidean n- sace, R n, the volume of the bodies and of their Minkowski sum K+L = {x+y : x K and y L}, are related by V (K + L) n V (K) n + V (L) n, with equality if and only if K and L are homothetic. As the first milestone of the Brunn-Minkowski theory, the Brunn-Minkowski inequality is a far-reaching generalization of the isoerimetric inequality. The Brunn-Minkowski inequality exoses the crucial log-concavity roerty of the volume functional because the Brunn-Minkowski inequality has an equivalent formulation as: for all real λ [0, 1], (1.1) V ((1 λ)k + λl) V (K) 1 λ V (L) λ, and for λ (0, 1), there is equality if and only if K and L are translates. A big art of the classical Brunn-Minkowski theory is concerned with establishing generalizations and analogues of the Brunn-Minkowski inequality for other geometric invariants. The excellent survey article of Gardner [16] gives a comrehensive account of various asects and consequences of the Brunn- Minkowski inequality. If h K and h L are the suort functions (see (2.1) for the definition) of K and L, the Minkowski combination (1 λ)k + λl is given by an intersection of half-saces, (1 λ)k + λl = {x R n : x u (1 λ)h K (u) + λ}, u S n 1 where x u denotes the standard inner roduct of x and u in R n. Assume that K and L are convex bodies that contain the origin in their interiors, then the geometric Minkowski combination, (1 λ) K+ o λ L, is defined by (1.2) (1 λ) K+ o λ L = u S n 1 {x R n : x u h K (u) 1 λ λ }. Date: February 25, Mathematics Subject Classification. 52A40. Key words and hrases. Brunn-Minkowski inequality, Brunn-Minkowski-Firey inequality, Minkowski mixedvolume inequality, Minkowski-Firey L -combinations. Research suorted, in art, by NSF Grant DMS

2 2 The arithmetic-geometric-mean inequality shows that for convex bodies K, L and λ [0, 1], (1.3) (1 λ) K+ o λ L (1 λ)k + λl. What makes the geometric Minkowski combinations difficult to work with is that while the convex body (1 λ)k +λl has (1 λ)h K +λh L as its suort function, the convex body (1 λ) K+ o λ L is the Wulff shae of the function h 1 λ K hλ L. The authors conjecture that for origin-symmetric bodies (i.e., unit balls of finite dimensional Banach saces), there is a stronger inequality than the Brunn-Minkowski inequality (1.1), the log-brunn-minkowski inequality: Problem 1.1. Show that if K and L are origin-symmetric convex bodies in R n, then for all λ [0, 1], (1.4) V ((1 λ) K+ o λ L) V (K) 1 λ V (L) λ. That the log-brunn-minkowski inequality (1.4) is stronger than its classical counterart (1.1) can be seen from the arithmetic-geometric mean inequality (1.3). Simle examles (e.g. an origincentered cube and one of its translates) shows that (1.4) cannot hold for all convex bodies. As is well known, the classical Brunn-Minkowski inequality (1.1) has as a consequence an inequality of fundamental imortance: the Minkowski mixed-volume inequality. One of the aims of this aer is to show that the log-brunn-minkowski inequality (1.4) also has an imortant consequence, the log-minkowski inequality: Problem 1.2. Show that if K and L are origin-symmetric convex bodies in R n, then (1.5) log h L d S h V K 1 V (L) log n 1 K n V (K). Here V K is the cone-volume robability measure of K (see definitions (2.5), (2.6), (2.8)). Just as the log-brunn-minkowski inequality (1.4) is stronger than its classical counterart (1.1), the log-minkowski inequality (1.5) turns out to be stronger than its classical counterart. The classical Minkowski mixed-volume inequality and the classical Brunn-Minkowski inequality are equivalent in that once either of these inequalities has been established, then the other can be obtained as a simle consequence. One of the aims of this aer is to demonstrate that the log-brunn-minkowski inequality (1.4) and the log-minkowski inequality (1.5) are equivalent in that once either of these inequalities has been established, then the other can be obtained as a simle consequence. Even in the lane the above roblems are non-trivial and unsolved. One of the aims of this aer is to establish the lane log-brunn-minkowski inequality along with its equality conditions: Theorem 1.3. If K and L are origin-symmetric convex bodies in the lane, then for all λ [0, 1], (1.6) V ((1 λ) K+ o λ L) V (K) 1 λ V (L) λ. When λ (0, 1), equality in the inequality holds if and only if K and L are dilates or K and L are arallelograms with arallel sides. In addition, in the lane, we will establish the log-minkowski inequality along with its equality conditions: Theorem 1.4. If K and L are origin-symmetric convex bodies in the lane, then, (1.7) log h L d S h V K 1 V (L) log 1 K 2 V (K), with equality if and only if, either K and L are dilates or K and L are arallelograms with arallel sides.

3 THE LOG-BRUNN-MINKOWSKI INEQUALITY 3 The above Minkowski combinations and roblems are merely two (imortant) frames of a long film. In the early 1960 s, Firey (see e.g. Schneider [54,. 383]) defined for each 1, what have become known as Minkowski-Firey L -combinations (or simly L -combinations) of convex bodies. If K and L are convex bodies that contain the origin in their interiors and 0 λ 1 then the Minkowski-Firey L -combination, (1 λ) K+ λ L, is defined by (1.8) (1 λ) K+ λ L = u S n 1 {x R n : x u ((1 λ)h K (u) + λ ) 1/ }. Firey also established the L -Brunn-Minkowski inequality (also known as the Brunn-Minkowski- Firey inequality): If > 1, then (1.9) V ((1 λ) K+ λ L) V (K) 1 λ V (L) λ, with equality for λ (0, 1) if and only if K = L. In the mid 1990 s, it was shown in [35, 36], that a study of the volume of Minkowski-Firey L -combinations leads to an embryonic L -Brunn- Minkowski theory. This theory has exanded raidly. (See e.g. [4, 7 9, 16, 18 25, 27 50, 52, 55 57, 59 61].) Note that definition (1.8) makes sense for all > 0. The case where = 0 is the limiting case given by (1.2). The crucial difference between the cases where 0 < < 1 and the cases where 1 is that the function ((1 λ)h K + λh L )1/ is the suort function of (1 λ) K+ λ L when 1, but it is not whenever 0 < < 1. When 0 < < 1, the convex body (1 λ) K+ λ L is the Wulff shae of ((1 λ)h K + λh L )1/. Unfortunately, rogress in the L -Brunn-Minkowski theory for < 1 has been slow. The resent work is a ste in that direction. It is easily seen from definition (1.8) that for fixed convex bodies K, L and fixed λ [0, 1], the L -Minkowski-Firey combination (1 λ) K+ λ L is increasing with resect to set inclusion, as increases; i.e., if 0 q, (1.10) (1 λ) K+ λ L (1 λ) K+ q λ L. From (1.10) one sees that the classical Brunn-Minkowski inequality (1.1) (i.e. the case = 1 of (1.9)) immediately yields Firey s L -Brunn-Minkowski inequality (1.9) for each > 1. The difficult situation arises when [0, 1) because now we are seeking inequalities that are stronger than the classical Brunn-Minkowski inequality. The L -Brunn-Minkowski inequality (1.9) cannot be established for all convex bodies that contain the origins in their interiors, for any fixed < 1. Even an origin-centered cube and one of its translates show that. However, the following roblem is of fundamental imortance in the L -Brunn-Minkowski theory: Problem 1.5. Suose 0 < < 1. Show that if K and L are origin-symmetric convex bodies in R n, then for all λ [0, 1], (1.11) V ((1 λ) K+ λ L) V (K) 1 λ V (L) λ. From the monotonicity of the L -Minkowski combination (1.10), it is clear that the log-brunn- Minkowski inequality imlies the L -Brunn-Minkowski inequalities for each > 0. We note that there are easy examles that show that the L -Brunn-Minkowski inequality (1.11) fails to hold for any < 0 even if attention were restricted to simle origin symmetric bodies. One of the aims of this aer is to show that the L -Brunn-Minkowski inequality (1.5) can be formulated equivalently as the L -Minkowski inequality:

4 4 Problem 1.6. Suose 0 < < 1. Show that if K and L are origin-symmetric convex bodies in R n, then ( ( ) ) 1 ( ) 1 hl (1.12) d V V (L) n K. V (K) S n 1 h K For each 1, the inequalities (1.11) and (1.12) are well known to hold for all convex bodies (that contain the origin in their interior) and are also well known to be equivalent, in that given one, the other is an easy consequence. From Jensen s inequality it can be seen that the L -Minkowski inequality (1.12) for the case = 0, the log-minkowski inequality (1.5), is the strongest of the L -Minkowski inequalities (1.12). The L -Minkowski inequality for the case = 1, the classical Minkowski mixed-volume inequality, is weaker than all the cases of (1.12) where (0, 1). Even in the lane the above roblems are non-trivial and unsolved. One of the aims of this aer is to solve the roblems in the lane. Solutions in higher dimensions would be highly desirable. We will rove the following theorems. Theorem 1.7. Suose 0 < < 1. If K and L are origin-symmetric convex bodies in the lane, then for all λ [0, 1], (1.13) V ((1 λ) K+ λ L) V (K) 1 λ V (L) λ. When λ (0, 1), equality in the inequality holds if and only if K = L. Observe that the equality conditions here are different than those of Theorem 1.3. Theorem 1.8. Suose 0 < < 1. If K and L are origin-symmetric convex bodies in the lane, then, ( ( ) ) 1 ( ) 1 hl (1.14) d V V (L) 2 K, V (K) S 1 h K with equality if and only if K and L are dilates. Observe that the equality conditions here are different than those of Theorem 1.4. The aroach used in this aer to establish the geometric inequalities of these theorems is new. 2. reliminaries For quick later reference we develo some notation and basic facts about convex bodies. Good general references for the theory of convex bodies are rovided by the books of Gardner [15], Gruber [17], Schneider [54], and Thomson [58]. The suort function h K : R n R, of a comact, convex set K R n is defined, for x R n, by (2.1) h K (x) = max{x y : y K}, and uniquely determines the convex set. Obviously, for a air K, L R n of comact, convex sets, we have (2.2) h k h L, if and only if, K L. Note that suort functions are ositively homogeneous of degree one and subadditive. A convex body is a comact convex subset of R n with non-emty interior. A boundary oint x K of the convex body K is said to have u S n 1 as one of its outer unit normals rovided x u = h K (u). A boundary oint is said to be singular if it has more than one unit normal vector. It is well known (see, e.g., [54]) that the set of singular boundary oints of a convex body has (n 1)-dimensional Hausdorff measure H n 1 equal to 0.

5 THE LOG-BRUNN-MINKOWSKI INEQUALITY 5 Let K be a convex body in R n and ν K : K S n 1 the generalized Gauss ma. For arbitrary convex bodies, the generalized Gauss ma is roerly defined as a ma into subsets of S n 1. However, H n 1 -almost everywhere on K it can be defined as a ma into S n 1. For each Borel set ω S n 1, the inverse sherical image ν 1 K (ω) of ω is the set of all boundary oints of K which have an outer unit normal belonging to the set ω. Associated with each convex body K in R n is a Borel measure S K on S n 1 called the Aleksandrov-Fenchel-Jessen surface area measure of K, defined by (2.3) S K (ω) = H n 1 (ν 1 K (ω)), for each Borel set ω S n 1 ; i.e., S K (ω) is the (n 1)-dimensional Hausdorff measure of the set of all oints on K that have a unit normal that lies in ω. The set of convex bodies will be viewed as equied with the Hausdorff metric and thus a sequence of convex bodies, K i, is said to converge to a body K, i.e., lim K i = K, i rovided that their suort functions converge in C(S n 1 ), with resect to the max-norm, i.e., h Ki h K 0. We shall make use of the weak continuity of surface area measures; i.e., if K is a convex body and K i is a sequence of convex bodies then (2.4) lim i K i = K = lim i S Ki = S K, weakly. Let K be a convex body in R n that contains the origin in its interior. The cone-volume measure V K of K is a Borel measure on the unit shere S n 1 defined for a Borel ω S n 1 by (2.5) V K (ω) = 1 x ν K (x) dh n 1 (x), n x ν 1 K (ω) and thus (2.6) dv K = 1 n h K ds K. Since, (2.7) V (K) = 1 n u S n 1 h K (u) ds K (u), we can turn the cone-volume measure into a robability measure on the unit shere by normalizing it by the volume of the body. The cone-volume robability measure V K of K is defined (2.8) VK = 1 V (K) V K. Suose K, L are convex bodies in R n that contain the origin in their interiors. For 0, the L -mixed volume V (K, L) can be defined as ( ) hl (2.9) V (K, L) = dv K. We need the normalized L -mixed volume V (K, L), which was first defined in [43], ( ) 1 ( ( ) V (K, L) ) 1 hl V (K, L) = = d V (K) V K. S n 1 h K S n 1 h K

6 6 Letting 0 gives ( V 0 (K, L) = ex log h L S n 1 d h V K K which is the normalized log-mixed volume of K and L. Obviously, from Jensen s inequality we know that V (K, L) is strictly monotone increasing, unless h L /h K is constant on sus K. Suose that the function k t (u) = k(t, u) : I S n 1 (0, ) is continuous, where I R is an interval. For fixed t I, let K t = ), u S n 1 {x R n : x u k(t, u)} be the Wulff shae (or Aleksandrov body) associated with the function k t. We shall make use of the well-known fact that (2.10) h Kt k t and h Kt = k t, a.e. w.r.t. S Kt, for each t I. If k t haens to be the suort function of a convex body then h Kt = k t, everywhere. The following lemma (roved in e.g. [23]) will be needed. Lemma 2.1. Suose k(t, u) : I S n 1 (0, ) is continuous, where I R is an oen interval. Suose also that the convergence in k(t, u) t = lim s 0 k(t + s, u) k(t, u) s is uniform on S n 1. If {K t } t I is the family of Wulff shaes associated with k t, then dv (K t ) k(t, u) = ds Kt (u). dt S t n 1 Suose K, L are convex bodies in R n. The inradius r(k, L) and outradius R(K, L) of K with resect to L are defined by r(k, L) = su{t > 0 : x + tl K and x R n }, R(K, L) = inf{t > 0 : x + tl K and x R n }. If L is the unit ball, then r(k, L) and R(K, L) are the radii of maximal inscribable and minimal circumscribable balls of K, resectively. Obviously from the definition, it follows that (2.11) r(k, L) = 1/R(L, K). If K, L haen to be origin-symmetric convex bodies, then obviously (2.12) r(k, L) = min u S n 1 h K (u) and R(K, L) = max u S n 1 h K (u). It will be convenient to always translate K so that for 0 t < r = r(k, L), the function k t = h K th L is strictly ositive. Let K t denote the Wulff shae associated with the function k t ; i.e., let K t be the convex body given by (2.13) K t = {x R n : x u h K (u) t for all u S n 1 }. Note that K 0 = K, and that obviously lim K t = K 0 = K. t 0 From definition (2.13) and (2.2) we immediately have (2.14) K t = {x R n : x + tl K}.

7 THE LOG-BRUNN-MINKOWSKI INEQUALITY 7 Using (2.14) we can extend the definition of K t for the case where t = r = r(k, L): K r = {x R n : x + rl K}. It is not hard to show (see e.g. the roof of (6.5.11) in [54]) that K r is a degenerate convex set (i.e. has emty interior) and that (2.15) lim t r V (K t ) = V (K r ) = 0. From Lemma 2.1 and (2.9), we obtain the well-known fact that for 0 < t < r = r(k, L), d (2.16) dt V (K t) = nv 1 (K t, L). Integrating both sides of (2.16), and using (2.15), gives Lemma 2.2. Suose K and L are convex bodies, and for 0 t < r = r(k, L), the body K t is the Wulff shae associated with the ositive function k t = h K th L. Then, for 0 t r = r(k, L), (2.17) V (K) V (K t ) = n where K r = {x R n : x + rl K}. t 0 V 1 (K s, L) ds, 3. Equivalence of the L -Brunn-Minkowski and the L -Minkowski inequalities In this section, we show that for each fixed 0 the L -Brunn-Minkowski inequality and the L -Minkowski inequality are equivalent in that one is an easy consequence of the other. In articular, the log-brunn-minkowski inequality and the log-minkowski inequality are equivalent. Suose > 0. If K and L are convex bodies that contain the origin and s, t 0 (not both zero) the L -Minkowski combination s K+ t L, is defined by s K+ t L = {x R n : x u (sh K (u) + t ) 1/ for all u S n 1 }. We see that for a convex body K and real s 0 the relationshi between the L -scalar multilication, s K, and Minkowski scalar multilication sk is given by: s K = s 1 K. Suose > 0 is fixed and suose the following weak L -BrunnMinkowski inequality holds for all origin-symmetric convex bodies K and L in R n such that V (K) = 1 = V (L): (3.1) V ((1 λ) K+ λ L) 1, for all λ (0, 1). We claim that from this it follows that the following seemingly stronger L -Brunn-Minkowski inequality holds: If K and L are origin-symmetric convex bodies in R n, then (3.2) V (s K+ t L) n sv (K) n + tv (L) n, for all s, t 0. To see this assume that the weak L -Brunn-Minkowski inequality (3.1) holds and that K and L are arbitrary origin-symmetric convex bodies. Let K = V (K) 1 n K and L = V (L) 1 n L. Then (3.1) gives (3.3) V ((1 λ) K+ λ L) 1. Let λ = V (L) n V (K) n +V (L) n. Then (1 λ) K+ λ L = 1 (K+ (V (K) n + V (L) n ) 1 L).

8 8 Therefore, from (3.3), we get V (K+ L) n V (K) n + V (L) n. If we now relace K by s K and L by t L and note that V (s K) n desired stronger L -Brunn-Minkowski inequality (3.2). = sv (K) n, we obtain the Lemma 3.1. Suose > 0. For origin symmetric convex bodies in R n, the L -Brunn-Minkowski inequality (1.11) and the L -Minkowski inequality (1.12) are equivalent. Proof. Suose K and L are fixed origin-symmetric convex bodies in R n. For 0 λ 1, let Q λ = (1 λ) K+ λ L; i.e., Q λ is the Wulff shae associated with the function q λ = ((1 λ)h K + λh L ) 1. It will be convenient to consider q λ as being defined for λ in the oen interval ( ɛ o, 1 + ɛ o ), where ɛ o > 0 is chosen so that for λ ( ɛ o, 1 + ɛ o ), the function q λ is strictly ositive. We first assume that the L -Minkowski inequality (1.12) holds. From (2.7), the fact that h Qλ = ((1 λ)h K + λh L ) 1 a.e. with resect to the surface area measure SQλ, (2.6) and (2.9), and finally the L -Minkowski inequality (1.12), we have (3.4) This gives, V (Q λ ) = 1 h Qλ ds Qλ n S n 1 = 1 ((1 λ)h K n + λh L )h1 Q λ S n 1 = (1 λ)v (Q λ, K) + λv (Q λ, L) (1 λ)v (Q λ ) n n V (K) ds Qλ n + λv (Qλ ) n n V (L) n. (3.5) V (Q λ ) ((1 λ)v (K) n + λv (L) n ) n/ V (K) 1 λ V (L) λ, which is the L -Brunn-Minkowski inequality (1.11). Now assume that the L -Brunn-Minkowski inequality (1.11) holds. As was seen at the beginning of this section, this inequality (in fact a seemingly weaker one) imlies the seemingly stronger L - Brunn-Minkowski inequality (3.2). But this inequality tells us that the function f : [0, 1] (0, ), given by f(λ) = V (Q λ ) n for λ [0, 1], is concave. The convex body Q λ is the Wulff shae of the function q λ = ((1 λ) h K + λ h L )1/. Now, the convergence as λ 0 in q λ q 0 λ h1 K (h L h K ) = h1 K h L h K, is uniform on S n 1. By Lemma 2.1, (2.6) and (2.9), and (2.7), dv (Q λ ) 1 K dλ = h L h K ds K = n λ=0 [V (K, L) V (K)]. Therefore, the concavity of f yields S n 1 h V (K) n n (V (K, L) V (K)) = f (0) f(1) f(0) = V (L) n V (K) n, which gives the L -Minkowski inequality (1.12). Lemma 3.2. For origin symmetric convex bodies in R n, the log-brunn-minkowski inequality (1.4) and the log-minkowski inequality (1.5) are equivalent.

9 THE LOG-BRUNN-MINKOWSKI INEQUALITY 9 Proof. Suose K and L are fixed origin-symmetric convex bodies in R n. For 0 λ 1, let Q λ = (1 λ) K+ o λ L; i.e., Q λ is the Wulff shae associated with the function q λ = h 1 λ K hλ L. It will be convenient to consider q λ as being defined for all λ in the oen interval ( ɛ o, 1 + ɛ o ), for some sufficiently small ɛ o > 0 and let Q λ be the Wulff shae associated with the function q λ. Observe that since q 0 and q 1 are the suort functions of convex bodies, Q 0 = K and Q 1 = L. First suose that we have the log-minkowski inequality (1.5) for K and L. Now h Qλ = h 1 λ K hλ L a.e. with resect to S Qλ, and thus, (3.6) 1 nv (Q λ ) Sn 1 h Qλ log h1 λ K hλ L ds Qλ 0 = h Qλ 1 = (1 λ) h Qλ log h K 1 ds Qλ + λ nv (Q λ ) S h n 1 Qλ nv (Q λ ) (1 λ) 1 V (K) log n V (Q λ ) + λ 1 V (L) log n V (Q λ ) = 1 n log V (K)1 λ V (L) λ. V (Q λ ) S n 1 h Qλ log h L h Qλ ds Qλ This gives the log-brunn-minkowski inequality (1.4). Suose now that we have the log-brunn-minkowski inequality (1.4) for K and L. The body Q λ is the Wulff shae associated wit the function q λ = h 1 λ K hλ L, and the convergence as λ 0 in q λ q 0 h K log h L, λ h K is uniform on S n 1. By Lemma 2.1, (3.7) dv (Q λ ) dλ = λ=0 S n 1 h K log h L h K ds K. But the log-brunn-minkowski inequality (1.4) tells us that λ log V (Q λ ) is a concave function, and thus 1 dv (Q λ ) (3.8) V (Q 0 ) dλ V (Q 1 ) V (Q 0 ) = log V (L) log V (K). λ=0 When (3.7) and (3.8) are combined the result is the log-minkowski inequality (1.5). 4. Blaschke s extension of the Bonnesen inequality From this oint forward we shall work exclusively in the Euclidean lane. We will make use of the roerties of mixed-volumes of comact convex sets, some of which might ossibly be degenerate (i.e. lower-dimensional). For quick later reference we list these roerties now. Suose K, L are lane comact convex sets. Of fundamental imortance is the fact that for real s, t 0, the area, V (sk + tl), of the Minkowski linear combination sk + tl = {sx + ty : x K and y L} is a homogeneous olynomial of degree 2 in s and t: (4.1) V (sk + tl) = s 2 V (K) + 2stV (K, L) + t 2 V (L). The coefficient V (K, L), the mixed area of K and L, is uniquely defined by (4.1) if we require (as we always will) it to be symmetric in its arguments; i.e. (4.2) V (K, L) = V (L, K).

10 10 From its definition, we see that the mixed area functional V (, ) is obviously invariant under indeendent translations of its arguments. Obviously, for each K, (4.3) V (K, K) = V (K). The mixed area of K, L is just the mixed volume V 1 (K, L) in the lane and thus (from (2.9) we see) it has the integral reresentation (4.4) V (K, L) = 1 ds K (u). 2 S 1 If K is degenerate with K = {su : c s c}, where u S 1 and c > 0, then S K is an even measure concentrated on the two oint set {±u } with total mass 4c. From (4.1), or from (4.4), we see that for lane comact convex K, L, L and real s, s 0, (4.5) V (K, sl + s L ) = sv (K, L) + s V (K, L ). But this, together with (4.2), shows that the mixed area functional V (, ) is linear with resect to Minkowski linear combinations in both arguments. From (4.4) we see that for lane comact convex K, L, L, we have (4.6) L L = V (K,L) V (K, L ), with equality if and only if h L = h L a.e. w.r.t. S K The basic inequality in this section, inequality (4.7), is Blaschke s extension of the Bonnesen inequality. It was roved using integral geometric techniques. It has been a valuable tool used to establish various isoerimetric inequalities, see e.g., [5], [6], [12], [51], and [53]. Since the equality conditions of inequality (4.7) are one of the critical ingredients in the roof of the log-brunn- Minkowski inequality, we resent a comlete roof of inequality (4.7), with its equality conditions. Theorem 4.1. If K, L are lane convex bodies, then for r(k, L) t R(K, L), (4.7) V (K) 2tV (K, L) + t 2 V (L) 0. The inequality is strict whenever r(k, L) < t < R(K, L). When t = r(k, L), equality will occur in (4.7) if and only if K is the Minkowski sum of a dilation of L and a line segment. When t = R(K, L), equality will occur in (4.7) if and only if L is the Minkowski sum of a dilation of K and a line segment. Proof. Let r = r(k, L) and suose t [0, r]. Recall from (2.13) that and that from (2.14), we have K t = {x R n : x u h K (u) t for all u S n 1 }, (4.8) K t + tl K. But (4.8), together with the monotonicity (4.6), linearity (4.5), and symmetry (4.2) of mixed volumes, together with (4.3) gives (4.9) V (K, L) V (K t + tl, L) = V (K t, L) + tv (L). Now Lemma 2.2 and (4.9) gives, (4.10) V (K) V (K t ) = 2 2 t 0 t 0 V (K s, L) ds (V (K, L) sv (L)) ds = 2tV (K, L) t 2 V (L).

11 THE LOG-BRUNN-MINKOWSKI INEQUALITY 11 Thus, (4.11) V (K) 2tV (K, L) + t 2 V (L) V (K t ). From (4.9) and (4.10) we see that equality holds in (4.11) if and only if, (4.12) V (K, L) = V (K s + sl, L), for all s [0, t], which, from (4.8) and (4.6), gives h K = h Ks + sh L, a.e. w.r.t. S L for all s [0, t]. By (2.15) we know V (K r ) = 0 and thus K r is a line segment, ossibly a single oint. Therefore, from (4.11) we have (4.13) V (K) 2rV (K, L) + r 2 V (L) 0. We will now establish the equality conditions in (4.13). To that end, suose: (4.14) V (K) 2rV (K, L) + r 2 V (L) = 0. Then by (4.12) we have, But this in (4.14) gives: which, using (4.5), can be rewritten as V (K, L) = V (K r + rl, L). V (K) 2rV (K r + rl, L) + r 2 V (L) = 0, V (K) 2rV (K r, L) r 2 V (L) = 0, and since V (K r ) = 0 can be written, using (4.5), as V (K) V (K r + rl) = 0. Since K r + rl K, the equality of their volumes forces us to conclude that in fact K r + rl = K. Therefore, K is the Minkowski sum of a dilation of L and the line segment K r (which may be a oint). Since 1/R(K, L) = r(l, K) from (2.12), from inequality (4.13), and its established equality conditions, we get V (L) 2r V (L, K) + r 2 V (K) 0, where r = r(l, K) = 1/R(K, L), with equality if and only if L is the Minkowski sum of a dilation of K and a line segment. But, using the symmetry of mixd volumes (4.2), this means that (4.15) V (K) 2RV (K, L) + R 2 V (L) 0, where R = R(K, L), with equality if and only if L is the Minkowski sum of a dilation of K and a line segment. Finally, inequalities (4.13) and (4.15) together with the well-known roerties of quadratic functions show that V (K) 2tV (K, L) + t 2 V (L) < 0, whenever r(k, L) < t < R(K, L).

12 12 5. Uniqueness of lanar cone-volume measure Given a finite Borel measure on the unit shere, under what necessary and sufficient conditions is the measure the cone-volume measure of a convex body? This is the unsolved log-minkowski roblem. It requires solving a Monge-Amère equation and is connected with some imortant curvature flows (see e.g. [2], [14], [56]). Uniqueness for the log-minkowski roblem is more difficult than existence. Even in the lane, the uniqueness of cone volume measure has not been settled. If the cone-volume measure is that of a smooth origin-symmetric convex body that has ositive curvature, uniqueness for lane convex bodies was established by Gage [14] and in the case of even, discrete, measures in the lane is treated by Stancu [56]. In this section, we shall establish the uniqueness of cone-volume measure for arbitrary symmetric lane convex bodies. For non-symmetric lane convex bodies the roblem remains both oen and imortant. The uniqueness of cone-volume measure is related to Firey s worn stone roblem. In determining the ultimate shae of a worn stone, Firey [11] showed that if the cone-volume measure of a smooth origin-symmetric convex body in R n is a constant multile of the Lebesgue measure (on S n 1 ), then the convex body must be a ball. This established uniqueness for the worn stone roblem for the symmetric case. In R 3, Andrews [2] established the uniqueness of solutions to the worn stone roblem by showing that a smooth (not necessarily symmetric) convex body in R 3 must be a ball if its cone volume measure is a constant multile of Lebesgue measure on S 2. The following inequality (5.1) was established by Gage [14] when the convex bodies are smooth and of ositive curvature. A limit rocess gives the general case, but the equality conditions do not follow. As will be seen, the equality conditions are critical for establishing the uniqueness of cone-volume measures in the lane. Lemma 5.1. If K, L are origin-symmetric lane convex bodies, then h 2 K (5.1) ds K V (K) h L ds K, S h 1 L V (L) S 1 with equality if and only if K and L are dilates, or K and L are arallelograms with arallel sides. Proof. Since K and L are origin symmetric, from (2.12) we have r(k, L) h K(u) for all u S 1. Thus, from Theorem 4.1 we get S 1 V (K) 2 h K(u) V (K, L) + R(K, L), ( ) 2 hk (u) V (L) 0. Integrating both sides of this, with resect to the measure h L ds K, and using (4.4) and (2.7), gives ( 0 V (K) 2 h ( ) 2 K(u) hk (u) V (K, L) + V (L)) ds K (u) = 2V (K)V (K, L) + V (L) This yields the desired inequality (5.1). Suose there is equality in (5.1). Thus, (5.2) V (K) 2 h K(u) V (K, L) + S1 h K (u) 2 ( hk (u) ds K(u). ) 2 V (L) = 0, for all u su S K.

13 THE LOG-BRUNN-MINKOWSKI INEQUALITY 13 If K and L are dilates, we re done. So assume that K and L are not dilates. But K L imlies that r(k, L) < R(K, L). From Theorem 4.1, we know that when it follows that and thus we conclude that r(k, L) < h K(u) V (K) 2 h K(u) V (K, L) + < R(K, L), ( ) 2 hk (u) V (L) < 0, (5.3) h K (u)/ {r(k, L), R(K, L)} for all u su S K. Note that since K is origin symmetric su S K is origin symmetric as well. Either there exists u 0 su S K so that h K (u 0 )/h L (u 0 ) = r(k, L) or h K (u 0 )/h L (u 0 ) = R(K, L). Suose that h K (u 0 )/h L (u 0 ) = r(k, L). Then from (5.2) and the equality conditions of Theorem 4.1 we know that K must be a dilation of the Minkowski sum of L and a line segment. But K and L are not dilaltes, so there exists an x 0 0 so that h K (u) = x 0 u + r(k, L), for all unit vectors u. This together with h K (u 0 )/h L (u 0 ) = r(k, L) shows that x 0 is orthogonal to u 0 and that the only unit vectors at which h K /h L = r(k, L) are u 0 and u 0. But su S K must contain at least one unit vector u 1 su S K other than ±u 0. From (5.3), and the fact that the only unit vectors at which h K /h L = r(k, L) are u 0 and u 0, we conclude h K (u 1 )/h L (u 1 ) = R(K, L) and by the same argument we conclude that the only unit vectors at which h K /h L = R(K, L) are u 1 and u 1. Now (5.3) allows us to conclude that su S K = {±u 0, ±u 1 }. This imlies that K is a arallelogram. Since K is the Minkowski sum of a dilate of L and a line segment, L must be a arallelogram with sides arallel to those of K. If we had assumed that h K (u 0 )/h L (u 0 ) = R(K, L), rather than r(k, L), the same argument would lead to the same conclusion. It is easily seen that the equality holds in (5.1) if K and L are dilates. A trivial calculation shows that equality holds in (5.1) if K and L are arallelograms with arallel sides. The following theorem was established by Gage [14] when the convex bodies are smooth and have ositive curvature. When the convex bodies are olytoes it is due to Stancu [57]. Theorem 5.2. If K and L are lane origin-symmetric convex bodies that have the same conevolume measure, then either K = L or else K and L are arallelograms with arallel sides. Proof. Assume that K L. Since V K = V L, it follows that V (K) = V (L). Thus, since K L, the bodies cannot be dilates. Thus inequality (5.1) becomes h L h K h K h L (5.4) dv K dv K and dv L dv L, S h 1 K S h 1 L S h 1 L S h 1 K

14 14 with equality, in either inequality, if and only if K and L are arallelograms with arallel sides. Using (5.4) and the fact that V K = V L, both twice, we get S h 1 K (u) dv h K (u) K(u) S h 1 L (u) dv K(u) h K (u) = S h 1 L (u) dv L(u) S h 1 K (u) dv L(u) = h K (u) dv K(u). Thus, we have equality in both inequalities of (5.4) and from the equality conditions of (5.4) we conclude that K and L are arallelograms with arallel sides. 6. Minimizing the logarithmic mixed volume Lemma 6.1. Suose K is a lane origin-symmetric convex body, with V (K) = 1, that is not a arallelogram. Suose also that P k is an unbounded sequence of origin-symmetric arallelograms all of which have orthogonal diagonals, and and such that V (P k ) 2. Then, the sequence S 1 log h Pk (u) dv K (u) is not bounded from above. Proof. Let u 1,k, u 2,k be orthogonal unit vectors along the diagonals of P k. Denote the vertices of P k by ±h 1,k u 1,k, ±h 2,k u 2,k. Without loss of generality, assume that 0 < h 1,k h 2,k. The condition V (P k ) 2 is equivalent to h 1,k h 2,k 1. The suort function of P k is given by (6.1) h Pk (u) = max{h 1,k u u 1,k, h 2,k u u 2,k }, for u S 1. Since S 1 is comact, the sequences u 1,k and u 2,k have convergent subsequences. Again, without loss of generality, we may assume that the sequences u 1,k and u 2,k are themselves convergent with lim 1,k = u 1 k and lim u 2,k = u 2, k where u 1 and u 2 are orthogonal. It is easy to see that if the cone-volume measure, V K ({±u 1 }), of the two-oint set {±u 1 } is ositive, then K contains a arallelogram whose area is 2V K ({±u 1 }). Since K itself is not a arallelogram and V (K) = 1, it must be the case that (6.2) V K ({±u 1 }) < 1 2. For δ (0, 1 3 ), consider the neighborhood, U δ, of {±u 1 }, on S 1, S 1 U δ = {u S 1 : u u 1 > 1 δ}. Since V K (S 1 ) = V (K) = 1, we see that for all or δ (0, 1 3 ) (6.3) V K (U δ ) + V K (U c δ ) = 1, where U c δ is the comlement of U δ. Since the U δ are decreasing (with resect to set inclusion) in δ and have a limit of {±u 1 }, lim V K(U δ ) = V K ({±u 1 }). δ 0 +

15 THE LOG-BRUNN-MINKOWSKI INEQUALITY 15 This together with (6.2), shows the existence of a δ o > 0 such that V K (U δo ) < 1 2. But this imlies that there is a small ɛ o (0, 1 ) so that 2 (6.4) τ o = V K (U δo ) ɛ o < 0. This together with (6.3) gives (6.5) V K (U δo ) = 1 2 ɛ o + τ o and V K (U c δ o ) = ɛ o τ o. Since u ik converge to u i, we have, u ik u i < δ o whenever k is sufficiently large (for both k = 1 and k = 2). Then for u U δo and k sufficiently large, we have u u 1,k u u 1 u (u 1,k u 1 ) u u 1 u 1,k u 1 1 δ o δ o δ o, where the last inequality follows from the fact that δ o < 1 3. For all u S1, we know that u u u u 2 2 = 1. Thus, for u U c δ o, we have u u 2 > (1 (1 δ o ) 2 ) 1 2 > 2δ o, which shows that when k is sufficiently large, u u 2,k u u 2 u (u 2,k u 2 ) u u 2 u 2,k u 2 2δ o δ o = δ o. From the last aragrah and (6.1) it follows that when k is sufficiently large, { δ o h 1,k if u U δo, (6.6) h Pk (u) δ o h 2,k if u Uδ c o. By (6.6) and (6.3), (6.5), the fact that 0 < h 1,k h 2,k together with (6.4), and finally the fact that h 1,k h 2,k 1 together with ɛ o (0, 1 ), we see that for sufficiently large k, 3 log h Pk dv K = log h Pk dv K + log h Pk dv K S 1 U δo U c δo log δ o + V K (U δo ) log h 1,k + V K (U c δ o ) log h 2,k = log δ o + ( τ o ɛ o ) log h 1,k + ( 1 2 τ o + ɛ o ) log h 2,k = log δ o + 2ɛ o log h 2,k + ( 1 2 ɛ o) log(h 1,k h 2,k ) + τ o (log h 1,k log h 2,k ) log δ o + 2ɛ o log h 2,k. Since P k is not bounded, the sequence h 2,k is not bounded from above. Thus, the sequence is not bounded from above. S 1 log h Pk dv K

16 16 Lemma 6.2. If K is a lane origin-symmetric convex body that is not a arallelogram, then there exists a lane origin-symmetric convex body K 0 so that V (K 0 ) = 1 and log h Q dv K log h K0 dv K S 1 S 1 for every lane origin-symmetric convex body Q with V(Q)=1. Proof. Obviously, we may assume that V (K) = 1. Consider the minimization roblem, inf log h Q dv K S 1 where the infimum is taken over all lane origin-symmetric convex bodies Q with V (Q) = 1. Suose that Q k is a minimizing sequence; i.e., Q k is a sequence of origin-symmetric convex bodies with V (Q k ) = 1 and such that log h S 1 Qk dv K tends to the infimum (which may be ). We shall show that the sequence Q k is bounded and the infimum is finite. By John s Theorem, there exist ellises E k centered at the origin so that (6.7) E k Q k 2E k. Let u 1,k, u 2,k, be the rincial directions of E k so that h 1,k h 2,k, where h 1,k = h Ek (u 1,k ) and h 2,k = h Ek (u 2,k ). Let P k be the origin-centered arallelogram that has vertices {±h 1,k u 1,k, ±h 2,k u 2,k }. (Observe that by the Princial Axis Theorem the diagonals of P k are erendicular.) Because of E k 2P k, it follows from (6.7) that (6.8) P k Q k 2P k. From this and V (Q k ) = 1, we see that V (P k ) 1. 4 Assume that Q k is not bounded. Then P k is not bounded. Alying Lemma 6.1 to 8P k shows that the sequence log h S 1 Pk dv K is not bounded from above. Therefore, from (6.8) we see that the sequence log h S 1 Qk dv K cannot be bounded from above. But this is imossible because Q k was chosen to be a minimizing sequence. We conclude that Q k is bounded. By the Blaschke Selection Theorem, Q k has a convergent subsequence that converges to an origin-symmetric convex body K 0, with V (K 0 ) = 1. It follows that log h S 1 K0 dv K is the desired infimum. We reeat the statement of Theorem 1.4: 7. The log-minkowski inequality Theorem 7.1. If K and L are lane origin-symmetric convex bodies, then log h L d S h V K 1 V (L) log 1 K 2 V (K), with equality if and only if either K and L are dilates or when K and L are arallelograms with arallel sides. Proof. Without loss of generality, we can assume that V (K) = V (L) = 1. We shall establish the theorem by roving log h L dv K S 1 log h K dv K, S 1 with equality if and only if either K and L are dilates or if they are arallelograms with arallel sides.

17 THE LOG-BRUNN-MINKOWSKI INEQUALITY 17 First, assume that K is not a arallelogram. Consider the minimization roblem min log h Q dv K, S 1 taken over all lane origin-symmetric convex bodies Q with V (Q) = 1. Let K 0 denote a solution, whose existence is guaranteed by Lemma 6.2. (Our aim is to rove that K 0 = K and thereby demonstrate that K itself can be the only solution to this minimization roblem.) Suose f is an arbitrary but fixed even continuous function. For some sufficiently small δ o > 0, consider the deformation of h K0, defined on ( δ o, δ o ) S 1, by q t (u) = q(t, u) = h K0 (u)e tf(u). Let Q t be the Wulff shae associated with q t. Observe that Q t is an origin symmetric convex body and that since q 0 is the suort function of the convex body K 0, we have Q 0 = K 0. Since K 0 is an assumed solution of the minimization roblem, the function defined on ( δ o, δ o ) by t V (Q t ) 1 2 ex{ S 1 log h Qt dv K }, attains a minimal value at t = 0. Since h Qt function defined on ( δ o, δ o ) by q t this function is dominated by the differentiable t V (Q t ) 1 2 ex{ S 1 log q t dv K }. But clearly both functions have the same value at 0 and thus the latter function attains a local minimum at 0. Thus, differentiating the latter function at t = 0, by using Lemma 2.1, and recalling that V (Q 0 ) = V (K 0 ) = 1, shows that 1 h K0 (u)f(u) ds K0 (u) + f(u) dv K (u) = 0. 2 S 1 S 1 Thus, since f was an arbitrary even function, we conclude that f(u) dv K0 (u) = f(u) dv K (u) S 1 S 1 for every even continuous f, and therefore, V K = V K0. By Theorem 5.2, and the assumtion that K is not a arallelogram, we conclude that K 0 = K. Thus, for each L such that V (L) = 1, log h L dv K log h K dv K, S 1 S 1 with equality if and only if K = L. This is the desired result when K is not a arallelogram. If K is a arallelogram the roof is trivial, but for the sake of comleteness we shall include it. Assume that K is the arallelogram whose suort function, for u S 1, is given by h K (u) = a 1 v 1 u + a 2 v 2 u, where v 1, v 2 S 1 and a 1, a 2 > 0. Then sus K = {±v1, ±v2 }, while V K ({±vi }) = 2a 1 a 2 v 1 v2, and v 1 v2 = v 2 v1. It is easily seen that V (K) = 4a 1 a 2 v 1 v2 = 1, and that (7.1) ex log h L dv K = h L (v1 )h L (v2 ). S 1

18 18 Recall that V (L) = 1. The arallelogram circumscribed about L with sides arallel to those of K has volume 4h L (v 1 )h L (v 2 ) v 1 v 2 1 = 16a 1 a 2 h L (v 1 )h L (v 2 ), and thus, 16a 1 a 2 h L (v 1 )h L (v 2 ) V (L) = 1, or equivalently h L (v 1 )h L (v 2 ) 1 16a 1 a 2, with equality if and only if L itself is a arallelogram with sides arallel to those of K. Thus, by (7.1), the functional S 1 log h L dv K attains its minimal value if and only if h L (v1 )h L (v2 1 ) = ; 16a 1 a 2 i.e., if and only if L is a arallelogram with sides arallel to those of K. Proof of Theorem 1.3. Lemma 3.2 shows that the log-minkowski inequality of Theorem 7.1 yields the log-brunn-minkowski inequality (1.6) of Theorem 1.3. To obtain the equality conditions of the log-brunn-minkowski inequality (1.6), we need to analyze the equality conditions of the inequality (3.6) in the roof of Lemma 3.2. The equality conditions for the log-minkowski inequality of Theorem 7.1 show that equality in inequality (3.6) would imly that either K, L and Q λ are dilates or that K, L and Q λ are arallelograms with arallel sides. This establishes the equality conditions of Theorem 1.3. Proof of Theorem 1.8. Jensen s inequality (along with its equality conditions), shows that the L - Minkowski inequality, for > 0, of Theorem 1.8 follows from the L 0 -Minkowski inequality of Theorem 7.1. Proof of Theorem 1.7. Lemma 3.1 shows that the L -Minkowski inequality of Theorem 1.8 yields the L -Brunn-Minkowski inequality of Theorem 1.7. To obtain the equality conditions of the L -Brunn-Minkowski inequality (1.13) of Theorem 1.7 we need to analyze the equality conditions of inequalities (3.4) and (3.5) of Lemma 3.1 which were used to derive the L -Brunn-Minkowski inequality of Theorem 1.7 from the L -Minkowski inequality of Theorem 1.8. From the equality conditions of Theorem 1.8, we know that equality in inequality (3.4) imlies that K and L are dilates. But inequality (3.5) is a direct consequence of the concavity of the log function and this concavity is strict. Hence, equality in inequality (3.5) imlies that V (K) = V (L). Thus we conclude that equality in the L -Brunn-Minkowski inequality (1.13) of Theorem 1.7 imlies that K = L. Acknowledgement The authors thank Professor Rolf Schneider for helful comments on an earlier draft of this aer. References [1] A.D. Alexandrov, Existence and uniqueness of a convex surface with a given integral curvature, Doklady Acad. Nauk Kasah SSSR 36 (1942), , MR , Zbl [2] B. Andrews, Gauss curvature flow: the fate of the rolling stones, Invent. Math. 138 (1999), , MR , Zbl [3] B. Andrews, Classification of limiting shaes for isotroic curve flows, J. Amer. Math. Soc. 16 (2003), , MR , Zbl [4] J. Bastero and M. Romance, Positions of convex bodies associated to extremal roblems and isotroic measures, Adv. Math. 184 (2004), 64 88, MR , Zbl

19 THE LOG-BRUNN-MINKOWSKI INEQUALITY 19 [5] W. Blaschke, Integralgeometrie. XI: Zur Variationsrechnung, Abh. math. Sem. Hansische Univ. 11 (1936), , Zbl [6] W. Blaschke, Vorlesungen über Integralgeometrie. II, Teubner, Leizig, 1937; rerint, Chelsea, New York, 1949 [7] S. Cami and P. Gronchi, The L -Busemann-Petty centroid inequality, Adv. Math. 167 (2002), , MR , Zbl [8] A. Cianchi, E. Lutwak, D. Yang, and G. Zhang, Affine Moser-Trudinger and Morrey-Sobolev inequalities, Calc. Var. Partial Differential Equations 36 (2009), , MR , Zbl [9] K. S. Chou and X. J. Wang, The L -Minkowski roblem and the Minkowski roblem in centroaffine geometry, Adv. Math. 205 (2006), 33 83, MR , Zbl [10] N. Dafnis and G. Paouris, Small ball robability estimates, Ψ 2 -behavior and the hyerlane conjecture, J. Funct. Anal. 258 (2010), , MR , Zbl [11] Wm. J. Firey, Shaes of worn stones, Mathematika 21 (1974), 1 11, MR , Zbl [12] H. Flanders, A roof of Minkowski s inequality for convex curves, Amer. Math. Monthly 75 (1968), , MR , Zbl [13] B. Fleury, O. Guédon, and G. A. Paouris, A stability result for mean width of L -centroid bodies, Adv. Math. 214 (2007), , MR , Zbl [14] M.E. Gage, Evolving lane curves by curvature in relative geometries, Duke Math. J. 72 (1993), , MR , Zbl [15] R.J. Gardner, Geometric Tomograhy, 2nd edition, Encycloedia of Mathematics and its Alications, vol. 58, Cambridge University Press, Cambridge, 2006, MR , Zbl , [16] R.J. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. 39 (2002), , MR , Zbl [17] P.M. Gruber Convex and discrete geometry, Grundlehren der Mathematischen Wissenschaften, 336. Sringer, Berlin, 2007, MR , Zbl [18] C. Haberl, L intersection bodies, Adv. Math. 217 (2008) , MR , Zbl [19] C. Haberl, Star body valued valuations, Indiana Univ. Math. J. 58 (2009), , MR , Zbl [20] C. Haberl and M. Ludwig, A characterization of L intersection bodies, Int. Math. Res. Not. 2006, Article ID 10548, 29., MR , Zbl [21] C. Haberl and F. Schuster, General L affine isoerimetric inequalities, J. Differential Geom. 83 (2009), 1 26, MR , Zbl [22] C. Haberl and F. Schuster, Asymmetric affine L Sobolev inequalities, J. Funct. Anal. 257, 2009, , MR , Zbl [23] C. Haberl, E. Lutwak, D. Yang, and G. Zhang, The even Orlicz Minkowski roblem, Adv. Math. 224 (2010), , MR , Zbl [24] C. Hu, X.-N. Ma, and C. Shen, On the Christoffel-Minkowski roblem of Firey s -sum, Calc. Var. Partial Differential Equations 21 (2004), , MR , Zbl [25] D. Hug, E. Lutwak, D. Yang, and G. Zhang, On the L Minkowski roblem for olytoes, Discrete Comut. Geom. 33 (2005), , MR , Zbl [26] K. Leichtweiss, Affine geometry of convex bodies, Johann Ambrosius Barth Verlag, Heidelberg, 1998, MR , Zbl [27] M. Ludwig, Projection bodies and valuations, Adv. Math. 172 (2002), , MR , Zbl [28] M. Ludwig, Valuations on olytoes containing the origin in their interiors, Adv. Math. 170 (2002), , MR , Zbl [29] M. Ludwig, Ellisoids and matrix-valued valuations, Duke Math. J. 119 (2003), , MR , Zbl [30] M. Ludwig, Minkowski valuations, Trans. Amer. Math. Soc. 357 (2005), , MR , Zbl [31] M. Ludwig, Intersection bodies and valuations, Amer. J. Math. 128 (2006), , MR , Zbl [32] M. Ludwig, General affine surface areas, Adv. Math. 224 (2010), , MR , Zbl [33] M. Ludwig and M. Reitzner. A classification of SL(n) invariant valuations, Ann. of Math. 172 (2010), , MR , Zbl [34] M. Ludwig, J. Xiao, and G. Zhang, Shar convex Lorentz-Sobolev inequalities, Math. Ann. 350 (2011), , MR , Zbl

20 20 [35] E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski roblem, J. Differential Geom. 38 (1993), , MR , Zbl [36] E. Lutwak, The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas, Adv. Math. 118 (1996), , MR , Zbl [37] E. Lutwak and V. Oliker, On the regularity of solutions to a generalization of the Minkowski roblem, J. Differential Geom. 41 (1995), , MR , Zbl [38] E. Lutwak, D. Yang, and G. Zhang, L affine isoerimetric inequalities, J. Differential Geom. 56 (2000), , MR , Zbl [39] E. Lutwak, D. Yang, and G. Zhang, A new ellisoid associated with convex bodies, Duke Math. J. 104 (2000), , MR , Zbl [40] E. Lutwak, D. Yang, and G. Zhang, The Cramer-Rao inequality for star bodies, Duke Math. J. 112 (2002), 59 81, MR , Zbl [41] E. Lutwak, D. Yang, and G. Zhang, Shar affine L Sobolev inequalities, J. Differential Geom. 62 (2002), 17 38, MR , Zbl [42] E. Lutwak, D. Yang, and G. Zhang, Volume inequalities for subsaces of L, J. Differential Geom. 68 (2004), , MR , Zbl [43] E. Lutwak, D. Yang, and G. Zhang, L John ellisoids, Proc. London Math. Soc. 90 (2005), , MR , Zbl [44] E. Lutwak, D. Yang, and G. Zhang, Otimal Sobolev norms and the L Minkowski roblem, Int. Math. Res. Not. (2006), Article ID 62987, 1 21, MR , Zbl [45] E. Lutwak, D. Yang, and G. Zhang, Volume inequalities for isotroic measures, Amer. J. Math. 129 (2007), , MR , Zbl [46] E. Lutwak and G. Zhang, Blaschke-Santaló inequalities, J. Differential Geom. 47 (1997), 1 16, MR , Zbl [47] M. Meyer and E. Werner, On the -affine surface area, Adv. Math. 152 (2000), , MR , Zbl [48] G. Paouris, Concentration of mass on convex bodies, Geom. Funct. Anal. 16 (2006), , MR , Zbl [49] G. Paouris, Small ball robability estimates for log-concave measures Grigoris Paouris, Trans. Amer. Math. Soc. 364 (2012), , MR [50] G. Paouris and E. Werner, Relative entroy of cone measures and L centroid bodies, Proc. London Math. Soc, (in ress). [51] De-lin Ren, Toics in Integral Geometry, World Scientific, Singaore, 1994, MR , Zbl [52] D. Ryabogin and A. Zvavitch, The Fourier transform and Firey rojections of convex bodies, Indiana Univ. Math. J. 53 (2004), , MR , Zbl [53] L.A. Santaló, Integral geometry and geometric robability (Second edition), Cambridge Univ, Press, Cambridge, 2004, Zbl , MR [54] R. Schneider, Convex bodies: the Brunn-Minkowski theory, Encycloedia of Mathematics and its Alications, vol. 44, Cambridge University Press, Cambridge, 1993, MR , Zbl [55] C. Schütt and E. Werner, Surface bodies and -affine surface area, Adv. Math. 187 (2004), , MR , Zbl [56] A. Stancu, The discrete lanar L 0 -Minkowski roblem, Adv. Math. 167 (2002), , MR , Zbl [57] A. Stancu, On the number of solutions to the discrete two-dimensional L 0 -Minkowski roblem, Adv. Math. 180 (2003), , MR , Zbl [58] A. C. Thomson, Minkowski geometry, Encycloedia of Mathematics and its Alications, vol. 63, Cambridge University Press, Cambridge, 1996, MR , Zbl [59] E. Werner, On L -affine surface areas, Indiana Univ. Math. J. 56 (2007), , MR , Zbl [60] E. Werner and D.-P. Ye, New L affine isoerimetric inequalities, Adv. Math. 218 (2008), , MR , Zbl [61] V. Yaskin and M. Yaskina, Centroid bodies and comarison of volumes Indiana Univ. Math. J. 55 (2006), , MR , Zbl [62] G. Zhang, Geometric inequalities and inclusion measures of convex bodies, Mathematika 41 (1994), , MR , Zbl [63] G. Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999), , MR , Zbl