GENERAL MIXED CHORD-INTEGRALS OF STAR BODIES

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1 ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 46, Number 5, 206 GENERAL MIXED CHORD-INTEGRALS OF STAR BODIES YIBIN FENG AND WEIDONG WANG ABSTRACT. Mixed chord-integrals of star bodies were first defined by Lu [9]. In this paper, the concept of mixed chord-integrals is extended to general mixed chord-integrals, which is motivated by the recent wor on general L p -affine isoperimetric inequalities by Haberl, et al. [6]. For this new notion of general mixed chord-integrals, isoperimetric and Alesandrov-Fenchel type inequalities are established which generalize inequalities obtained by Lu, and a cyclic inequality is also obtained. Furthermore, we prove several Brunn-Minowsi type inequalities for general mixed chordintegrals.. Introduction and main results. Let S n denote the unit sphere in Euclidean space R n, and let V (K) denote the n-dimensional volume of a body K. For the standard unit ball B in R n, we write ω n = V (B) for its volume. If K is a compact star-shaped (about the origin) set in R n, then its radial function, ρ K = ρ(k, ) : R n \ {0} [0, ), is defined by (see, e.g., [8, 33]) ρ(k, u) = max{λ 0 : λu K}, u S n. If ρ K is positive and continuous, then K will be called a star body (about the origin), and S n denotes the set of star bodies in R n. We will use So n and Sc n to denote the subset of star bodies i n containing the origin in their interiors and whose centroids lie at the origin, respectively. Two star bodies, K and L, are said to be dilates of one another if ρ K (u)/ρ L (u) is independent of u S n. 200 AMS Mathematics subject classification. Primary 52A20, 52A40. Keywords and phrases. General mixed chord-integrals, mixed chord-integrals, star bodies. Research is supported by the Natural Science Foundation of China, grant No , and the Young Foundation of Hexi University, grant No. QN Received by the editors on December 6, 203, and in revised form on November 24, 204. DOI:0.26/RMJ Copyright c 206 Rocy Mountain Mathematics Consortium 499

2 500 YIBIN FENG AND WEIDONG WANG Lutwa [23] introduced the notion of mixed width-integrals of convex bodies. Motivated by Lutwa s ideas, the notion of mixed chord-integrals of star bodies was recently defined by Lu (see [9]): For K,..., K n So n, the mixed chord-integral, C(K,..., K n ), of K,..., K n was defined by (.) C(K,..., K n ) = c(k, u) c(k n, u0) ds(u), n where ds(u) is the (n )-dimensional volume element o n and c(k, u) denotes the half chord of K in the direction u, namely, c(k, u) = ρ(k, u)/2 + ρ(k, u)/2. Thus, the mixed chord-integral is a map So n So n R. It is positive, continuous and multilinear }{{} n with respect to radial Minowsi linear combinations, positively homogeneous and monotone under set inclusion. Star bodies K,..., K n are said to have a similar chord if there exist constants λ,..., λ n > 0 such that λ c(k, u) = = λ n c(k n, u) for all u S n. Lu [9] established the following isoperimetric and Alesandrov-Fenchel inequalities for mixed chord-integrals. Theorem.A. If K,..., K n S n o, then C(K,..., K n ) n V (K ) V (K n ), with equality if and only if K,..., K n are centered at the origin and dilates of each other. Theorem.B. If K,..., K n S n o and < m n, then m C(K,..., K n ) m C(K,..., K n m, K n i+,..., K n i+ ), i= with equality if and only if K n m+,..., K n are all of a similar chord. We now propose the definition of general mixed chord-integrals of star bodies, which generalizes the above definition of mixed chordintegrals.

3 MIXED CHORD-INTEGRALS OF STAR BODIES 50 For τ (, ), the general mixed chord-integral, (K,..., K n ), of K,..., K n So n is defined by (.2) (K,..., K n ) = c (K, u) c (K n, u) ds(u), n where c (K, u) = f ρ(k, u) + f 2 ρ(k, u) and the functions f and f 2 are defined as follows: (.3) f = From (.3), it immediately follows that ( + τ)2 2( + τ 2 ), f ( τ)2 2 = 2( + τ 2 ). (.4) f + f 2 = ; (.5) f ( τ) = f 2, f 2 ( τ) = f. By (.3), if we let τ = 0 in definition (.2), then C (0) (K,..., K n ) is just Lu s mixed chord-integral C(K,..., K n ). Similarly, the general mixed chord-integral, (K,..., K n ) : S n o S n o R, is also positive, continuous and multilinear with respect to the radial Minowsi linear combinations, positively homogeneous and monotone under set inclusion. Star bodies K,..., K n are said to have a similar general chord if there exist constants λ,..., λ n > 0 such that λ c (K, u) = = λ n c (K n, u) for all u S n. They are said to have a joint constant general chord if the product c (K, u) c (K n, u) is constant for all u S n. If we tae K = = K n i = K and K n i+ = = K n = B in definition (.2), then the general chord-integral of order i, i (K), of K So n is defined by (.6) i (K) = c (K, u) n i ds(u). n Taing K = = K n = K in (.2), we write (K) for (K,, K), and call it the general chord-integral of K S n o. The asymmetric (dual) Brunn-Minowsi theory has as its starting point the theory of valuations in connection with isoperimetric and analytic inequalities (see [, 6, 3, 4, 5, 6, 20, 2, 22, 27, 28, 29, 30, 3, 34, 36, 37, 38, 39, 40]).

4 502 YIBIN FENG AND WEIDONG WANG As our main results, we first establish extended versions of Theorem.A and Theorem.B, given by Theorem. and Theorem.2. Theorem.. If τ (, ) and K,..., K n S n o, then (.7) (K,, K n ) n V (K ) V (K n ), with equality if and only if K,..., K n are centered at the origin and dilates of each other. Theorem.2. If τ (, ) and K,..., K n So n, < m n, then (.8) m (K,..., K n ) m (K,..., K n m, K n i+,..., K n i+ ), i= with equality if and only if K n m+,..., K n are all of similar general chord. Moreover, we will prove the following cyclic inequality. Theorem.3. If τ (, ) and K, L S n o, then, for i < j <, (.9) i (K, L) j (K, L)j i j (K, L) i, with equality if and only if K and L have a similar general chord. Here, i (K, L) = (K, n i; L, i) in which K appears n i times and L appears i times. The proofs of Theorems..3 will be given iection 4 of this paper. Iection 3, we establish some properties of general chordintegrals of order i. Moreover, several Brunn-Minowsi type inequalities for general chord-integrals of order i are obtained iection Preliminaries. If E R n is non-empty, the polar set E of E is defined by (see [8]) E = {x R n : x y, for all y E}. For K S n c, an extension of the well-nown Blasche-Santaló inequality taes the following form (see [24]).

5 MIXED CHORD-INTEGRALS OF STAR BODIES 503 Theorem 2.A. If K S n c, then (2.) V (K)V (K ) ω 2 n, with equality if and only if K is an ellipsoid. If K,..., K m So n and λ,..., λ m R, then the radial Minowsi linear combination is defined by (see [25]) (2.2) ρ(λ K + +λ m K m, ) = λ ρ(k, ) + + λ m ρ(k m, ). For K, L So n and λ, µ 0 (both nonzero), the radial Blasche linear combination, λ K +µ L, of K and L is defined by (see [25]): (2.3) ρ(λ K +µ L, ) n = λρ(k, ) n + µρ(l, ) n. For more information on binary operations between convex or star bodies, we refer to the articles [9, 0, ]. For K So n, the intersection body of K, IK, is the star body symmetric with respect to origin whose radial function o n is given by (see [25]): (2.4) ρ(ik, u) = n S n u ρ(k, u) n dλ n 2 (u), where dλ n 2 (u) is (n 2)-dimensional spherical Lebesgue measure. For u S n, K u denotes the intersection of K with the subspace u that passes through the origin and is orthogonal to u. From equations (2.2), (2.3) and (2.4), it follows that, if K, L S n o and λ, µ 0 (both nonzero), then (2.5) I(λ K +µ L) = λik +µil. The polar coordinate formula for volume of a body K in R n is (2.6) V (K) = ρ(k, u) n ds(u). n 3. General chord-integrals of order i. In this section, we establish some properties and inequalities for general chord-integrals of order i.

6 504 YIBIN FENG AND WEIDONG WANG From definition (.2) and the multilinearity of general mixed chordintegrals, we see that the general chord-integral of λ K + +λ m K m is a homogeneous polynomial of degree n in λ,..., λ m. Theorem 3.. For τ (, ) and K,..., K m So n, let K = λ K + +λ m K m. Then m m (3.) (K) = λ j λ jn (K j,..., K jn ). j = j n= As a direct consequence of Theorem 3., we have: Theorem 3.2. For τ (, ) and K S n o, let K µ = K +µb (µ > 0). Then, for j = 0,,..., n, (3.2) j (K µ ) = n j ( n j i i=0 Lemma 3.3. If τ (, ) and K S n o, then (3.3) (K) V (K), ) j+i (K)µi. with equality if and only if K is centered at the origin. Proof. From Minowsi s inequality (see [7]), we have [ ] /n (K) /n = c (K, u) n ds(u) [ n = (f ρ(k, u) + f 2 ρ(k, u)) n ds(u) [ n ] /n (f ρ(k, u)) n ds(u) [ n ] /n + (f 2 ρ(k, u)) n ds(u) [ n ] /n = ρ(k, u) n ds(u), n ] /n

7 MIXED CHORD-INTEGRALS OF STAR BODIES 505 that is, (K) n S n ρ(k, u) n ds(u) = V (K). From the equality condition of Minowsi s inequality, we see that equality holds in (3.3) if and only if K and K are dilates of one another, i.e., K is centered at the origin. Theorem 3.4. If τ (, ) and K S n c, then, for 0 < i < n, (3.4) i (K) i (K ) ωn, 2 for i > n, inequality (3.4) is reversed, with equality in each inequality if and only if K is an ellipsoid centered at the origin. Proof. From Lemma 3.3 and Jensen s inequality (see [7]), we get for i > 0 and i n, [ /(n i) [ ] /n [ ] /n i (K)] (K) V (K), ω n ω n ω n i.e., (3.5) i (K) /(n i) ωn i/n(n i) V (K) /n. By (3.5), we have (3.6) i (K ) /(n i) ωn i/n(n i) V (K ) /n. Combining equations (3.5) and (3.6), it follows from the extended Blasche-Santaló inequality (see Theorem 2.A) that [ ] /(n i) (3.7) i (K) i (K ) ω 2/(n i) n. If 0 < i < n in inequality (3.7), then we have i (K) i (K ) ωn; 2 if i > n in inequality (3.7), then the above inequality is reversed. According to the equality conditions of inequality (2.), inequality (3.3) and Jensen s inequality, we see that equality holds in every inequality if and only if K is an ellipsoid centered at the origin.

8 506 YIBIN FENG AND WEIDONG WANG 4. Proofs of Theorems..3. Proof of Theorem.. By Hölder s inequality (see [7]), we have ( ) n (K,..., K n ) n = c (K, u) c (K n, u) ds(u) ( n ) (4.) c (K, u) n ds(u) ( n ) c (K n, u) n ds(u), n with equality if and only if K,, K n have similar general chord. An application of Minowsi s inequality (see [7]) thus yields (K,..., K n ) ( ) /n c (K, u) n ds(u) ( n ) /n c (K n, u) n ds(u) ( n ) /n = (f ρ(k, u) + f 2 ρ(k, u)) n ds(u) ( n ) /n (f ρ(k n, u) + f 2 ρ(k n, u)) n ds(u) ( n ) /n ρ(k, u) n ds(u) ( n ) /n ρ(k n, u) n ds(u) n = V (K ) /n V (K n ) /n, Conse- with equality in the second inequality if and only if τ = 0. quently, (K,, K n ) n V (K ) V (K n ). According to the equality conditions of the above inequalities, we see that equality holds in (.7) if and only if K,..., K n are dilates of each other and centered at the origin.

9 MIXED CHORD-INTEGRALS OF STAR BODIES 507 Lemma 4. ([26]). If f 0, f,..., f m are (strictly) positive continuous functions defined o n and λ,..., λ m are positive constants, the sum of whose reciprocals is unity, then (4.2) m [ /λi f 0 (u)f (u) f m (u) ds(u) f 0 (u)f λi i (u) ds(u)], S n S n with equality if and only if there exist positive constants α,..., α m such that α f λ (u) = = α mf λ m m (u) for all u S n. i= Proof of Theorem.2. In Lemma 4., we tae λ i = m ( i m); f 0 = c (K, u) c (K n m ), u f i = c (K n i+, u) Then it follows that ( i m). (f 0 = if m = n); c (K, u) c (K n, u) ds(u) S n m [ ] /m c (K, u) c (K n m, u)c (K n i+, u) m ds(u). S n i= By definition (.2), this yields (K,, K n ) m m (K,..., K n m, K n i+,..., K n i+ ). i= From the equality condition of inequality (4.2), we see that equality holds in (.8) if and only if K n m+,..., K n are all of a similar general chord. Proof of Theorem.3. From Hölder s inequality (see [7]), it follows that: i (K, L) ( j)/( i) (K, L)(j i)/( i) ( = c (K, u) n i c (L, u) i ds(u) n ) ( j)/( i)

10 508 YIBIN FENG AND WEIDONG WANG i.e., ( ) (j i)/( i) c (K, u) n c (L, u) ds(u) n c (K, u) n j c (L, u) j ds(u) = j (K, L), n i (K, L) j (K, L)j i j (K, L) i. From the equality condition of Hölder s inequality, we see that equality holds in (.9) if and only if K and L have a similar general chord. If i = 0, j = i and = n in inequality (.9), then we have the following fact. Corollary 4.2. If τ (, ) and K, L S n o, then, for 0 i n, (4.3) i (K, L) n (K) n i (L) i, for i < 0 or i > n, inequality (4.3) is reversed, with equality in every inequality if and only if i = n or, when i n, K and L have a similar general chord. If we tae i = and i = in Corollary 4.2, then we get the following versions of the dual Minowsi inequalities for general mixed chord-integrals. Corollary 4.3. If τ (, ) and K, L S n o, then (K, L) n (K) n (L), with equality if and only if K and L have a similar general chord. Corollary 4.4. If τ (, ) and K, L S n o, then (K, L)n (K) n+ (L), with equality if and only if K and L have a similar general chord.

11 MIXED CHORD-INTEGRALS OF STAR BODIES Brunn-Minowsi type inequalities. This section is dedicated to the study of Brunn-Minowsi inequalities for general chordintegrals of order i. Log-concavity properties of geometric functionals have long played an important role in analysis and geometry (see [2, 4, 32, 35, 4, 42] for some recent results). We first establish the following Brunn-Minowsi type inequality for general chord-integrals of order i with respect to the radial Minowsi addition. Theorem 5.. If τ (, ) and K, L S n o, then, for i n, (5.) i and, for i > n, (5.2) i (K +L) /(n i) i (K) /(n i) + i (L) /(n i) ; (K +L) /(n i) i (K) /(n i) + i (L) /(n i), with equality in each inequality if and only if K and L have a similar general chord. Proof. We first prove inequality (5.). For i n, it follows from Minowsi s inequality (see [7]) that ( ) /(n i) i (K +L) /(n i) = c (K +L, u) n i ds(u) ( n ) /(n i) = (c (K, u) + c (L, u)) n i ds(u) ( n ) /(n i) + n ( n This gives inequality (5.). inequality (5.2). c (K, u) n i ds(u) S n c (L, u) n i ds(u) S n = i (K) /(n i) + i (L) /(n i). ) /(n i) Similarly, Minowsi s inequality yields From the equality conditions of Minowsi s inequality, we see that equality holds in inequalities (5.) and (5.2) if and only if K and L have a similar general chord.

12 50 YIBIN FENG AND WEIDONG WANG Theorem 5.2. If K, L So n i n j n, (5.3) ( ) /(j i) ( C i (I(K +L)) C i j (I(K +L)) for j n i n, (5.4) ( C i (I(K +L)) j (I(K +L)) j ) /(j i) ( C i j and τ (, ), i, j R, then, for (IK) (IK) (IK) (IK) ) /(j i) + ( C i j ) /(j i) + ( C i j ) /(j i) (IL) ; (IL) ) /(j i) (IL), (IL) with equality in each inequality if and only if IK and IL have a similar general chord. In order to prove Theorem 5.2, the following lemmas are required. An extension of Becenbach s inequality (see [3]) was obtained by Dresher [5] through the means of moment-space techniques. Lemma 5.3 (The Becenbach-Dresher inequality). If p r 0, f, g 0, and ϕ is a distribution function, then (5.5) ( E(f + ) g)p dϕ /(p r) ( E (f + E f p ) dϕ /(p r) ( ) g)r dϕ E f + E gp dϕ /(p r), r dϕ E gr dϕ with equality if and only if the functions f and g are positively proportional. Here E is a bounded measurable subset in R n. Moreover, the inverse Becenbach-Dresher inequality was obtained by Li and Zhao [8]. Lemma 5.4 (The inverse Becenbach-Dresher inequality). If r 0 p, f, g 0, and ϕ is a distribution function, then (5.6) ( E(f + ) g)p dϕ /(p r) ( E (f + E f p ) dϕ /(p r) ( ) g)r dϕ E f + E gp dϕ /(p r), r dϕ E gr dϕ

13 MIXED CHORD-INTEGRALS OF STAR BODIES 5 with equality if and only if the functions f and g are positively proportional. Proof of Theorem 5.2. From equations (.2) and (2.5), it follows that, for p r 0, C n p(i(k +L)) = c (I(K +L), u) p ds(u) n = (5.7) c (IK +IL, u) p ds(u) n = (c (IK, u) + c (IL, u)) p ds(u). n Similarly, (5.8) n r(i(k +L)) = n S n (c (IK, u) + c (IL, u)) r ds(u). Combining equations (5.7) and (5.8), we obtain, using Lemma 5.3, (5.9) ( C n p(i(k +L)) ) /(p r) n r(i(k +L)) ( (c (IK, u) + c (IL, u)) p ds(u) = Sn (c S (IK, u) + c (IL, u)) r ds(u) ( n c (IK, u) p ) ds(u) /(p r) Sn c S (IK, u) r ds(u) ( n c (IL, u) p ) ds(u) /(p r) + Sn c S (IL, u) r ds(u) n ( ) C /(p r) ( ) n p(ik) C /(p r) n p(il) = +. C n r(ik) C n r(il) ) /(p r) Suppose p = n i and r = n j. It follows from 0 r p that i n j n. Taing p = n i and r = n j in (5.9), this yields the desired inequality (5.3). Using the same method, and Lemma 5.4 instead, we obtain inequality (5.4). From the equality conditions of inequalities (5.5) and (5.6), we see that equality holds in inequalities (5.3) and (5.4) if and only if

14 52 YIBIN FENG AND WEIDONG WANG c (IK, u) and c (IL, u) are positively proportional, that is, IK and IL have a similar general chord. If j = n in (5.3), then C n (I(K +L)) = C n (IK) = C n (IL) = ω n is a constant, and we obtain the following result. Corollary 5.5. If τ (, ) and K, L S n o, then, for i n, i (I(K +L)) /(n i) i (IK) /(n i) + i (IL) /(n i), with equality if and only if IK and IL have similar general chords. An inequality of Giannopoulos et al. [2] states that, if K is a convex body and L is an n-ball in R n, then, for = 0,..., n, (5.0) W (K + L) W + (K + L) W (K) W + (K) + W (L) W + (L). However, inequality (5.0) does not hold for an arbitrary pair of nonempty compact convex sets K and L. Also, (5.0) only holds if = n 2 or = n [7]. In the following, we will prove two analogous inequalities for general chord-integrals of order i. Theorem 5.6. If τ (, ), and K, L is an arbitrary pair of star bodies in R n, then, for = n 2 or = n, (5.) (K +L) + (K +L) C (K) + (K) + C (L) + (L). Proof. Let = n 2, and let B = (B,..., B) be an (n 2)-tuple of the unit ball B. It follows from Theorem.2 that, for all t, s 0, (K +sb, L +tb, B) 2 n 2 (K +sb) n 2 (L +tb) 0. Since general mixed chord-integrals are multilinear with respect to the radial [ Minowsi linear combination, we obtain [ ] s 2 n (L)2 ω n n 2 ]+2st (L) ω n (K, L, B) n (K)C n (L) + t 2 [ n (K)2 ω n n 2 (K) ] + g(s, t) 0,

15 MIXED CHORD-INTEGRALS OF STAR BODIES 53 where g(s, t) is a linear function of s and t. From Theorem.2, we now that (5.2) and (5.3) n (K)2 ω n n 2 (K) 0 n (L)2 ω n n 2 (L) 0. Together with (5.2), it follows that either (5.4) ω n (K, L, B) n (K)C n (L) 0, or (5.5) [ ] 2 ω n (K, L, B) n (K)C n (L) [ ] [ ] n (K)2 ω n n 2 (K) n (L)2 ω n n 2 (L). Using Theorem.2 again, we obtain for s, t 0, (5.6) n (sk +tl) 2 ω n n 2 (sk +tl) 0. Using the multilinearity of general mixed chord-integrals we obtain from (5.6): [ ] (5.7) s 2 n (L)2 ω n n 2 (L) [ ] + 2st n (K)C n (L) ω n (K, L, B) + t 2 [ n (K)2 ω n n 2 (K) ] 0. By equations (5.2) and (5.4), we now that, if (5.7) holds, then the discriminant of the above quadratic form (5.7) is non-positive. Thus, equation (5.5) always holds. Now using equation (5.5), it follows from the arithmetic geometric means inequality that ω n (K, L, B) n (K)C n (L) [ ω n n 2 (K) C n (K)2] /2 [ ω n n 2 (L) C n (L)2] /2 2 n (L) n (K) [ ω n n 2 (K) C n (K)2]

16 54 YIBIN FENG AND WEIDONG WANG that is, + n (K) 2 = ω n 2 n (L) [ C n (K)C (L) n 2 n n (K)C n (L), [ ω n n 2 (L) C n (L)2] (K) + C n (K) (5.8) 2 (K, L, B) C n (K)C (L) n 2 n C n (L) n 2 (L) ] (K) + C n (K) n (L) C n 2 (L). By the multilinearity of general mixed chord-integrals and inequality (5.8), we infer n 2 (K +L) = n 2 (K) + C n 2 (L) + 2C (K, L, B) C n n 2 (K) + C n 2 (L) + (K)C (L) n 2 (K) n = + C n (K) C n 2 n (L) (L) n 2 (L) [ C n 2 (K) n (K) + C n (L) ] ( ) n (K) + C n (L), that is, n 2 (K +L) n (K +L) C n 2 (K) n (K) + C n 2 (L) n (L). n For the case = n, note that since C n (L) = ω n, Theorem 5.6 reduces to the inequality n (K +L) n (K) + C n (L), (K +L) = C n (K) = which holds for every pair of star bodies.

17 MIXED CHORD-INTEGRALS OF STAR BODIES 55 Theorem 5.7. Let K be a star body and L an n-ball in R n τ (, ). Then, for all = 0,..., n, and (5.9) (K +L) + (K +L) C (K) + (K) + C (L) + (L). Proof. Let L = tb for t 0 and define, for every = 0,,..., n, f (s) = (K +sb). From the multilinearity of general mixed chord-integrals, it follows that f (s + ε) = (K +sb +εb) = (K +sb) + ε(n ) + (K +sb) + O(ε 2 ) = f (s) + ε(n )f + (s) + O(ε 2 ). Therefore, f (s) = (n )f + (s). Apply Theorem.2 to get, for = 0,,..., n 2, i.e., + (K +sb) 2 (K +sb) +2 (K +sb), f + (s) 2 f (s)f +2 (s). Define (5.20) F (s) = f (s) f + (s) It follows that (5.2) thus, F (s) = f (s)f +(s) f (s)f + (s) f + (s) 2 ( = 0,,..., n 2). = f +(s) 2 + (n )(f + (s) 2 f (s)f +2 (s)) f + (s) 2 ; (5.22) F (t) F (0) + t.

18 56 YIBIN FENG AND WEIDONG WANG Since (5.23) (L) + (L) = C (tb) + (tb) = t, it follows from equations (5.20), (5.22) and (5.23) that, for = 0,,..., n 2, (K +L) + (K +L) C (K) + (K) + C (L) + (L). When = n, inequality (5.9) always holds as an equality. Acnowledgments. The referee of this paper proposed many very valuable comments and suggestions to improve the accuracy and readability of the original manuscript. We would lie to express our most sincere thans to the anonymous referee. REFERENCES. J. Abardia and A. Bernig, Projection bodies in complex vector space, Adv. Math. 227 (20), S. Aleser, A. Bernig and F.E. Schuster, Harmonic analysis of translation invariant valuations, Geom. Funct. Anal. 2 (20), E.F. Becenbach and R. Bellman, Inequalities, 2nd edition, Springer-Verlag, Berlin, K.J. Böröczy, E. Lutwa, D. Yang and G. Zhang, The log-brunn-minowsi inequality, Adv. Math. 23 (202), M. Dresher, Moment spaces and inequalities, Due Math. J. 20 (953), Y.B. Feng and W.D. Wang, General L p-harmonic Blasche bodies, P. Indian Math. Soc. 24 (204), M. Fradelizi, Some inequalities about mixed volumes, Israel J. Math. 35 (2003), R.J. Gardner, Geometric tomography, Second edition, Cambridge University Press, Cambridge, R.J. Gardner, D. Hug and W. Weil, Operations between sets in geometry, J. Europ. Math. Soc. (JEMS) 5 (203), , The Orlicz-Brunn-Minowsi theory: A general framewor, additions, and inequalities, J. Diff. Geom. 97 (204), R.J. Gardner, L. Parapatits and F.E. Schuster, A characterization of Blasche addition, Adv. Math. 254 (204),

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20 58 YIBIN FENG AND WEIDONG WANG 34. F.E. Schuster, Convolutions and multiplier transformations of convex bodies, Trans. Amer. Math. Soc. 359 (2007), , Crofton measures and Minowsi valuations, Due Math. J. 54 (200), F.E. Schuster and M. Weberndorfer, Volume inequalities for asymmetric Wulff shapes, J. Diff. Geom. 92 (202), W.D. Wang and Y.B. Feng, A general L p -version of Petty s affine projection inequality, Taiwan J. Math. 7 (203), W.D. Wang and T.Y. Ma, Asymmetric L p -difference bodies, Proc. Amer. Math. Soc. 42 (204), W.D. Wang and X.Y. Wan, Shephard type problems for general L p- projection bodies, Taiwan J. Math. 6 (202), M. Weberndorfer, Shadow systems of asymmetric L p-zonotopes, Adv. Math. 240 (203), C.J. Zhao, On polars of Blasche-Minowsi homomorphisms, Math. Scand. (202), B. Zhu, N. Li and J. Zhou, Brunn-Minowsi type inequalities for L p - moment bodies, Glasgow Math. J. 55 (203), School of Mathematics and Statistics, Hexi University, Zhangye, , China address: fengyibin00@63.com Department of Mathematics, China Three Gorges University, Yichang, , China address: wangwd722@63.com