The Dual Orlicz-Aleksandrov-Fenchel Inequality Becomes Lutwak’S Dual N Aleksandrov-Fenchel Inequality

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h(K, x) = max{x · y : y ∈ K}, for x ∈ Rn, deﬁned the support function h(K, x) of K. A nonempty closed convex set is uniquely determined by its support function. Lp addition and inequalities are the fundamental and core content in the Lp Brunn-Minkowski theory. For recent important results and more information from this theory, we refer to x[16] [20-23], [29], [31], [35-39], [42-43], [48-49], [51-52], x[59] and the references therein. In recent years, progress towards an Orlicz-Brunn-Minkowski theory, initiated by Lutwak, Yang and Zhang [40] and [41]. Gardner, Hug and Weil [11] introduced a corresponding addition and constructed a general framework for the Orlicz-Brunn-Minkowski theory, and made clear for the ﬁrst time the relation to Orlicz spaces and norms, and Orlicz-Minkowski and Brunn-Minkowski inequalities were established.

The Orlicz addition of convex bodies was also introduced in different ways and extend the Lp-Brunn- Minkowski inequality to the Orlicz-Brunn-Minkowski inequality (see [53]). The Orlicz centroid inequality for star bodies was introduced in [61] which is an extension from convex to star bodies. The other articles advance the theory can be found in literatures [19], [25], [27], [45] and [55]. The radial addition K+L of star sets (compact sets that is star-shaped at o and contains o) K and L can be defined by e ρ(K+L, ·)= ρ(K, ·)+ ρ(L, ·), where ρ(K, ·) denotes the radial function ofe star set K. The radial function of star set K is defined by

ρ(K,u) = max{c ≥ 0 : cu ∈ K}, for u ∈ Sn−1. The origin and history of the radial addition can be referred to [10, p. 235]. When ρ(K, ·) is positive and continuous, K will be called a star body. Let Sn denote the set of star bodies about the origin in Rn. The radial addition and volume are the core and essence of the classical dual Brunn- Minkowski theory and played an important role in theory (see, e.g., [3], [14], [17], [18], [26], [28], [32] and [51] for recent important contributions). Lutwak [34] introduced the concept of dual mixed volume laid the foundation of the dual Bruun-Minkowski theory. What is particularly important is that this theory plays a very important and key role in solving the Busemann-Petty problem in [8], [14], [33] and [47].

For any p 6= 0, the Lp-radial addition K+pL deﬁned by (see [9])

p p p ρ(K+peL, x) = ρ(K, x) + ρ(L, x) ,

n n for x ∈ R and K,L ∈ S (see [12]).e Obviously, when p = 1, the Lp-radial addition +p becomes the radial addition +. e

e 2 The Lp-harmonic radial addition was deﬁned by Lutwak [31]: If K,L are star bodies, the Lp-harmonic radial addition, deﬁned by −p −p −p ρ(K+pL, x) = ρ(K, x) + ρ(L, x) , (1.1)

n for p ≥ 1 and x ∈ R . The Lp-harmonicb radial addition of convex bodies was ﬁrst studied by Firey

[6]. The operation of the Lp-harmonic radial addition and the Lp-dual Minkowski, Brunn-Minkwski

inequalities are the basic concept and inequalities in the Lp-dual Brunn-Minkowski theory. The latest information and important results of this theory can be referred to [3], [5], [9], [12], [15], [17-18] and [28] and the references therein. A systematic investigation on the concepts of the addition for convex body and star body, we refer to [11], [12] and [15]. The theory has recently turned to a study extending from

Lp-dual Brunn-Minkowski theory to the Orlicz dual Brunn-Minkowski theory. The dual Orlicz-Brunn- Minkowski theory has also attracted the attention, see [13], [54], [56-58] and [60]. The Orlicz harmonic

radial addition K+φL of two star bodies K and L, deﬁned by (see [60])

ρ(K,u) ρ(L,u) bρ(K+ L,u)) = sup λ> 0 : φ + φ ≤ φ(1) , (1.2) φ λ λ where u ∈ Sn−1, and bφ : (0, ∞) → (0, ∞) is a convex and decreasing function such that φ(0) = ∞,

limt→∞ φ(t) = 0 and limt→0 φ(t) = ∞. let C denote the class of the convex and decreasing functions φ. −p Obviously, if φ(t) = t and p ≥ 1, then the Orlicz harmonic addition becomes the Lp-harmonic radial

addition. The Orlicz dual mixed volume, denoted by Vφ(K,L), deﬁned by

′ φr(1) V (K+φε · L) − V (K) e 1 ρ(L,u) n Vφ(K,L) := lim = φ ρ(K,u) dS(u), (1.3) n ε→0+ ε n n−1 ρ(K,u) ZS b e where K+φε · L is the Orlicz linear combination of K and L (see Section 3), and the right derivative of a real-valued function φ is denoted by φ′ . When φ(t) = t−p and p ≥ 1, the Orlicz dual mixed volume b r Vφ(K,L) becomes the Lp dual mixed volume V−p(K,L), deﬁned by (see [31])

e 1 e n+p −p V−p(K,L)= ρ(K,u) ρ(L,u) dS(u). (1.4) n n−1 ZS n e If K1,...,Kn ∈ S , the dual mixed volume of star bodies K1,...,Kn, V (K1,...,Kn) deﬁned by Lutwak (see [34]) 1 e V (K1,...,Kn)= ρ(K1,u) ··· ρ(Kn,u)dS(u). (1.5) n n−1 ZS n Lutwak’s dual Aleksandrov-Fenchele inequality (see [33]) is following: If K1, ··· ,Kn ∈ S and 1 ≤ r ≤ n, then r 1 V (K1, ··· ,Kn) ≤ V (Ki ...,Ki,Kr+1,...,Kn) r , i=1 Y with equality if and only ifeK1,...,Kr are all dilationse of each other.

3 As we all know, Lp-dual volume Vp(K,L) of star bodies K and L has been extended to the Orlicz space and becomes Orlicz dual volume V (K,L). However the dual mixed volumes V (K , ··· ,K ) has e φ 1 n not been extended to the Orlicz space, and this question becomes a diﬃcult research in convex geometry. e e The main reason for the diﬃculty is that it has not been extended to Lp space. In the paper, we solved the

difficulty. Our main aim is to generalize direct the classical dual mixed volumes V (K1, ··· ,Kn) and dual Aleksandrov-Fenchel inequality to the Orlicz space without passing through L space. Under the frame- p e work of dual Orlicz-Brunn-Minkowski theory, we introduce a new affine geometric quantity by calculating the first order Orlicz variation of the dual mixed volumes, and call it Orlicz multiple dual mixed volumes, n denoted by Vφ(K1, ··· ,Kn,Ln), which involves (n + 1) star bodies in R . The fundamental notions and conclusions of the dual mixed volume V (K , ··· ,K ) and the dual Minkowski, and Aleksandrov-Fenchel e 1 n inequalities are extended to an Orlicz setting. The related concepts and conclusions of L multiple dual e p mixed volume Vp(K1, ··· ,Kn,Ln) and Lp-Aleksandrov-Fenchel inequality are also derived. The new dual Orlicz-Aleksandrov-Fenchel inequality in special case yields the dual Aleksandrov-Fenchel inequality and e the Orlicz dual Minkowski inequality for the dual quermassintegrals, respectively. As application, a new dual Orlicz-Brunn-Minkowski inequality for the Orlicz harmonic radial addition is established, which implies the dual Orlicz-Brunn-Minkowski inequality for the dual quermassintegrals. Complying with the spirit of introduction of Aleksandrov, Fenchel and Jensen’s mixed quermassin-

tegrals, and introduction of Lutwak’s Lp-mixed quermassintegrals, we calculate the ﬁrst order Orlicz variational of the dual mixed volumes. In Section 4, we prove that the ﬁrst order Orlicz variation of the dual mixed volumes can be expressed as:

d 1 V (L1+φε · K1,K2, ··· ,Kn)= ′ · Vφ(L1,K1, ··· ,Kn), dε + φ (1) ε=0 r

ne b e where L1,K1, ··· ,Kn ∈ S , φ ∈ C and ε > 0. In the above ﬁrst order variational equation, we ﬁnd a new geometric quantity. Based on this, we extract the required geometric quantity, denoted by

Vφ(L1,K1, ··· ,Kn) and call it Orlicz multiple dual mixed volume Vφ(L1,K1, ··· ,Kn) of (n + 1) star bodies L ,K , ··· ,K , deﬁned by e 1 1 n e

′ d Vφ(L1,K1, ··· ,Kn) := φr(1) · V (L1+φε · K1,K2, ··· ,Kn). dε + ε=0

e e b We also prove the new aﬃne geometric quantity Vφ( L1,K1, ··· ,Kn) has an integral representation.

1 ρ(K1,u)e Vφ(L1,K1, ··· ,Kn)= φ ρ(L1,u)ρ(K2,u) ··· ρ(Kn,u)dS(u), (1.6) n n−1 ρ(L ,u) ZS 1 e When K1 = L and L1 = K2 = ··· = Kn = K, the Orlicz multiple dual mixed volume Vφ(L1,K1, ··· ,Kn) becomes the usual Orlicz dual mixed volume V (K,L). When φ(t) = t−p, p ≥ 1, Orlicz multiple φ e e 4 dual mixed volume Vφ(L1,K1, ··· ,Kn) becomes a new dual mixed volume in Lp place, denoted by

V− (K ,...,K ,L ), call it L multiple dual mixed volume. From (1.6), we have p 1 n n e p

e 1 −p 1+p V−p(L1,K1, ··· ,Kn)= ρ(K1,u) ρ(L1,u) ρ(K2,u) ··· ρ(Kn,u)dS(u). (1.7) n n−1 ZS e Putting L1 = K1 in (1.7), the Lp multiple dual mixed volume V−p(L1,K1, ··· ,Kn) becomes the usual dual

mixed volume V (K , ··· ,K ). Putting K = L and L = K = ··· = K = K in (1.7), V− (L ,K , ··· ,K ) 1 n 1 1 2 e n p 1 1 n becomes the L dual mixed volume V− (K,L). Putting K = L and L = K = ··· = K − = K and ep p 1 1 2 e n i K − = ··· = K = B in (1.7), V− (L ,K , ··· ,K ) becomes the harmonic mixed p-quermassintegral, n i+1 n pe 1 1 n W− (K,L), deﬁned by (see Section 2) p,i e

f 1 n−i+p −p W−p,i(K,L)= ρ(K,u) ρ(L,u) dS(u). (1.8) n n−1 ZS In Section 5, we establishf the following dual Orlicz-Aleksandrov-Fenchel inequality for the Orlicz n multiple dual mixed volume. If L1,K1, ··· ,Kn ∈ S , φ ∈C and 1 ≤ r ≤ n, then

r 1 r i=1 V (Ki ...,Ki,Kr+1,...,Kn) Vφ(L1,K1,K2, ··· ,Kn) ≥ V (L1,K2, ··· ,Kn) · φ . (1.9) V (L1,K2 ...,Kn) ! Q e e e If φ is strictly convex, equality holds if and only if L1,K1,...,Ker are all dilations of each other. When −p φ(t) = t , p = 1 and K1 = L1, the dual Orlicz-Aleksandrov-Fenchel inequality becomes Lutwak’s dual n Aleksandrov-Fenchel inequality. If K1, ··· ,Kn ∈ S and 1 ≤ r ≤ n, then

r 1 V (K1, ··· ,Kn) ≤ V (Ki ...,Ki,Kr+1,...,Kn) r , (1.10) i=1 Y e e −p with equality if and only if K1,...,Kr are all dilations of each other. When φ(t) = t , p ≥ 1, the

dual Orlicz-Aleksandrov-Fenchel inequality (1.9) becomes the following Lp dual Aleksandrov-Fenchel n inequality. If L1,K1, ··· ,Kn ∈ S , p ≥ 1 and 1 ≤ r ≤ n, then

p+1 r V (L1,K2, ··· ,Kn) p ≤ V (Ki ...,Ki,Kr+1,...,Kn) r . (1.11) V−p(L1,K1,K2, ··· ,Kn) i=1 e Y e If φ is strictly convex,e equality holds if and only if L1,K1,...,Kr are all dilations of each other. When

K1 = L and L1 = K2 = ··· = Kn = K, the Orlicz dual Aleksandrov-Fenchel inequality (1.9) becomes the following Orlicz dual Minkowski inequality established in [60]. If K,L ∈ Sn and φ ∈C, then

1 V (L) n V (K,L) ≥ V (K) · φ , (1.12) φ V (K) ! e with equality if and only if K and L are dilates. In fact, in Section 5, we show also the Orlicz Aleksandrov- Fenchel inequality (1.9) in special case yield also the following result. If K,L ∈ Sn,0 ≤ i

5 then 1/(n−i) Wi(L) Wφ,i(K,L) ≥ Wi(K)φ . (1.13)  Wi(K)!  f f f   If φ is strictly convex, equality holds if and only if K andfL are dilates. Here Wi(K) is the usually dual quermassintegral of K, and W (K,L) is the Orlicz dual mixed quermassintegral of K and L, deﬁned by φ,i f (see Section 4) f 1 ρ(L,u) n−i Wφ,i(K,L)= φ ρ(K,u) dS(u). (1.14) n n−1 ρ(K,u) ZS In Section 6, we establishf the following Orlicz dual Brunn-Minkowski type inequality. If L1,K1, ··· ,Kn ∈ Sn and φ ∈C, then

1 1 V (K ) ··· V (K ) n V (L )V (K ) ··· V (K ) n φ(1) ≥ φ 1 n + φ 1 2 n . (1.15) n n  V (K1+φL1,K2, ··· ,Kn) !   V (K1+φL1,K2, ··· ,Kn) !      If φ is strictly convex,e equalityb holds if and only if L1,K1,...,Ke n bare all dilations of each other. Putting

K1 = K,L1 = L and K2 = ··· = Kn = K1+φL1 in (1.15), it follows the Orlicz dual Brunn-Minkowski inequality established in [58]. If K,L ∈ Sn and φ ∈C b 1 1 V (K) n V (L) n φ(1) ≥ φ + φ . (1.16) V (K+ L) V (K+ L) φ ! φ !

If φ is strictly convex, equality holds if andb only if K and L are dilates.b In fact, in Section 6, we show also the Orlicz dual Brunn-Minkowski inequality (1.15) in special case yield also the following result. If K,L ∈ Sn, φ ∈C and 0 ≤ i

1 1 − − W (K) n i W (L) n i φ(1) ≥ φ i + φ i . (1.17)  Wi(K+φL)!   Wi(K+φL)!  f f     If φ is strictly convex, equality holdsf if andb only if K and L aref dilates.b

2 Preliminaries

The setting for this paper is n-dimensional Euclidean space Rn. A body in Rn is a compact set equal to the closure of its interior. For a compact set K ⊂ Rn, we write V (K) for the (n-dimensional) Lebesgue measure of K and call this the volume of K. The unit ball in Rn and its surface are denoted by B and Sn−1, respectively. Let Kn denote the class of nonempty compact convex subsets containing the origin in their interiors in Rn. Associated with a compact subset K of Rn, which is star-shaped with respect to the origin and contains the origin, its radial function is ρ(K, ·): Sn−1 → [0, ∞), deﬁned by

ρ(K,u) = max{λ ≥ 0 : λu ∈ K}.

6 Two star bodies K and L are dilates if ρ(K,u)/ρ(L,u) is independent of u ∈ Sn−1. If λ> 0, then

ρ(λK,u)= λρ(K,u).

From the deﬁnition of the radial function, it follows immediately that for A ∈ GL(n) the radial function of the image AK = {Ay : y ∈ K} of K is given by (see e.g. [10])

− ρ(AK,u)= ρ(K, A 1u), for all u ∈ Sn−1. Namely, the radial function is homogeneous of degree −1. Let δ˜ denote the radial Hausdorﬀ metric, as follows, if K,L ∈ Sn, then (see e.g. [46])

δ˜(K,L)= |ρ(K,u) − ρ(L,u)|∞.

2.1 Dual mixed volumes

The polar coordinate formula for volume of a compact set K is 1 V (K)= ρ(K,u)ndS(u). (2.1) n n−1 ZS

The ﬁrst dual mixed volume, V1(K,L), deﬁned by

e 1 V (K+ε · L) − V (K) V1(K,L)= lim , n ε→0+ ε e where K,L ∈ Sn. The integral representatione for ﬁrst dual mixed volume is proved: For K,L ∈ Sn,

1 n−1 V1(K,L)= ρ(K,u) ρ(L,u)dS(u). (2.2) n n−1 ZS The Minkowski inequality for ﬁrste dual mixed volume is the following: If K,L ∈ Sn, then

n n−1 V1(K,L) ≤ V (K) V (L),

n with equality if and only if K and L aree dilates. (see [33]) If K1,...,Kn ∈ S , the dual mixed volume

V (K1,...,Kn) is deﬁned by (see [34]) 1 e V (K1,...,Kn)= ρ(K1,u) ··· ρ(Kn,u)dS(u). (2.3) n n−1 ZS e If K1 = ··· = Kn−i = K,Kn−i+1 = ··· = Kn = L, the dual mixed volume V (K1,...,Kn) is written as V (K,L). If L = B, the dual mixed volume V (K,L)= V (K,B) is written as W (K) and called dual i i i e i quermassintegral of K. For K ∈ Sn and 0 ≤ i

1 n−i Wi(K)= ρ(K,u) dS(u). (2.4) n n−1 ZS f 7 If K1 = ··· = Kn−i−1 = K,Kn−i = ··· = Kn−1 = B and Kn = L, the dual mixed volume

V (K,...,K,B,...,B,L) is written as Wi(K,L) and called dual mixed quermassintegral of K and L. n−i−1 i Fore K,L ∈ Sn and 0 ≤ i

Wi(K+ε · L) − Wi(K) 1 n−i−1 Wi(K,L) = lim = ρ(K,u) ρ(L,u)dS(u). (2.5) ε→0+ ε n n−1 ZS f e f The fundamentalf inequality for dual mixed quermassintegral stated that: If K,L ∈ Sn and 0 ≤ i

1/(n−i) 1/(n−i) 1/(n−i) Wi(K+L) ≤ Wi(K) + Wi(L) , (2.7) with equality if and only if fK andeL are dilates.f f

2.2 Lp-dual mixed volume

The dual mixed volume V−1(K,L) of star bodies K and L is deﬁned by ([31])

e V (K) − V (K+ε · L) V−1(K,L) = lim , (2.8) ε→0+ ε b where + is the harmonic addition.e The following is a integral representation for the dual mixed volume

V−1(K,L): b 1 n+1 −1 V−1(K,L)= ρ(K,u) ρ(L,u) dS(u). (2.9) e n n−1 ZS The dual Minkowski inequalitye for the dual mixed volume states that

n n+1 −1 V−1(K,L) ≥ V (K) V (L) , (2.10) with equality if and only if K and Leare dilates. (see ([32])) The dual Brunn-Minkowski inequality for the harmonic addition states that

− − − V (K+L) 1/n ≥ V (K) 1/n + V (L) 1/n, (2.11)

with equality if and only if K and L bare dilates (This inequality is due to Firey [6]).

The Lp dual mixed volume V−p(K,L) of K and L is deﬁned by ([31])

e p V (K+pε · L) − V (K) V−p(K,L)= − lim , (2.12) n ε→0+ ε b e 8 where K,L ∈ Sn and p ≥ 1. n The following is an integral representation for the Lp dual mixed volume: For K,L ∈ S and p ≥ 1,

1 n+p −p V−p(K,L)= ρ(K,u) ρ(L,u) dS(u). (2.13) n n−1 ZS e n Lp-dual Minkowski and Brunn-Minkowski inequalities were established by Lutwak [31]: If K,L ∈ S and p ≥ 1, then n n+p −p V−p(K,L) ≥ V (K) V (L) , (2.14) with equality if and only if K and Leare dilates, and

−p/n −p/n −p/n V (K+pL) ≥ V (K) + V (L) , (2.15) with equality if and only if K and Lbare dilates.

2.3 Mixed p-harmonic quermassintegral

From (1.1), it is easy to see that if K,L ∈ Sn, 0 ≤ i

p W (K+ ε · L) − W (L) 1 − − − lim i p i = ρ(K.u)n i+pρ(L.u) pdS(u). (2.16) n − i ε→0+ ε n n−1 ZS f b f Let K,L ∈ Sn, 0 ≤ i

by W−p,i(K,L), deﬁned by (see [50])

f 1 n−i+p −p W−p,i(K,L)= ρ(K,u) ρ(L,u) dS(u). (2.17) n n−1 ZS f Obviously, when K = L, the p-harmonic quermassintegral W−p,i(K,L) becomes the dual quermassintegral W (K). The Minkowski and Brunn-Minkowski inequalities for the mixed p-harmonic quermassinte- i f gral are following (see [50]): If K,L ∈ Sn, 0 ≤ i

n−i n−i+p −p W−p,i(K,L) ≥ Wi(K) Wi(L) , (2.18)

with equality if and only if K andf L are dilates. IffK,L ∈ Sn,f 0 ≤ i

−p/(n−i) −p/(n−i) −p/(n−i) Wi(K+pL) ≥ Wi(K) + Wi(L) , (2.19)

with equality if and onlyf if K andb L are dilates.f f Inequality (2.19) is a Brunn-Minkowski type inequality for the p-harmonic addition. For diﬀerent variants of the classical Brunn-Minkowski inequalities we refer to [1], [2], [4], [43] and [47] and the references therein.

9 3 Orlicz harmonic linear combination

n Throughout the paper, the standard orthonormal basis for R will be {e1,...,en}. Let Cm, m ∈ N, denote the set of convex function φ : [0, ∞)m → (0, ∞) that are strictly decreasing in each variable and satisfy φ(0) = ∞. When m = 1, we shall write C instead of C1. Orlicz harmonic radial addition is defined below. n Definition 3.1 Let m ≥ 2, φ ∈Cm, Kj ∈ S and j =1,...,m, define the Orlicz harmonic addition of K1,...,Km, denoted by +φ(K1,...,Km), defined by ρ(K , x) ρ(K , x) ρ(+ (K ,...,Kb ), x) = sup λ> 0 : φ 1 ,..., m ≤ φ(1) , (3.1) φ 1 m λ λ n for x ∈ R . Equivalently,b the Orlicz multiple harmonic addition +φ(K1,...,Km) can be defined implicitly by ρ(K , x) ρ(K b, x) φ 1 ,..., m = φ(1), (3.2) ρ(+ (K ,...,K ), x) ρ(+ (K ,...,K ), x) φ 1 m φ 1 m for all x ∈ Rn. An important special case is obtained when b b m φ(x1,...,xm)= φ(xj ), j=1 X

for φ(t) ∈ Cm. We then write +φ(K1,...,Km) = K1+φ ··· +φKm. This means that K1+φ ··· +φKm is deﬁned either by b b b b b m ρ(K , x) ρ(K + ··· + K , x) = sup λ> 0 : φ j ≤ φ(1) , (3.3) 1 φ φ m  λ  j=1  X  b b for all x ∈ Rn, or by the corresponding special case of (3.1). From (3.3), it follows easy that

m ρ(K , x) φ j = φ(1), λ j=1 X if and only if

λ = ρ(K1+φ ··· +φKm, x). (3.4)

Next, deﬁne the Orlicz harmonic linear combinationb b on the case m = 2.

Deﬁnition 3.2 The Orlicz harmonic linear combination, denotes +φ(K,L,α,β), deﬁned by ρ(K, x) ρ(L, x) αφ + βφ b = φ(1), (3.5) ρ(+ (K,L,α,β), x) ρ(+ (K,L,α,β), x) φ φ for K,L ∈ Sn, x ∈ Rn and α, β ≥ 0 (not both zero). b b −p When φ(t) = t and p ≥ 1, then Orlicz harmonic linear combination +φ(K,L,α,β) changes to the

Lp-harmonic linear combination α · K+pβ · L. We shall write K+φε · L instead of +φ(K,L, 1,ε), for ε ≥ 0 b

b 10 b b and assume throughout that this is deﬁned by (3.5), where α = 1,β = ε and φ ∈ C. It is easy that

+φ(K,L, 1, 1) = K+φL.

4b Orlicz multipleb dual mixed volume

Let us introduce the Orlicz multiple dual mixed volume.

Deﬁnition 4.1 For φ ∈C, we deﬁne the Orlicz dual mixed volume Vφ(L1,K1, ··· ,Kn) by

1 ρ(K1,u) e Vφ(L1,K1, ··· ,Kn) =: φ ρ(L1,u)ρ(K2,u) ··· ρ(Kn,u)dS(u), n n−1 ρ(L ,u) ZS 1 e n for all L1,K1,...,Kn ∈ S . To derive this deﬁnition 4.1, we need the following lemmas. n Lemma 4.2 ([60]) If K1,L1 ∈ S and φ ∈C, then

L1+φε · K1 → L1 (4.1) as ε → 0+. b n Lemma 4.3 If L1,K1,...,Kn ∈ S and φ ∈C, then

d 1 ρ(K1,u) V (L1+φε · K1,K2, ··· ,Kn)= ′ φ ρ(L1,u)ρ(K2,u) ··· ρ(Kn,u)dS(u). dε + nφ (1) n−1 ρ(L ,u) ε=0 r ZS 1 (4.2) e b n n−1 Proof Suppose ε> 0, K1,L1 ∈ S and u ∈ S , let

ρε = ρ(L1+φε · K1,u).

Since b ρ(L ,u) − ρ(K ,u) 1 = φ 1 φ(1) − εφ 1 , ρ ρ ε ε and from Lemma 4.2, and noting that φ is continuous function, we obtain

(ρ − ρ(L ,u))ρ(K ,u) ··· ρ(K ,u) lim ε 1 2 n ε→0+ ε

ρε ρε − ρ(L1,u) = ρ(K2,u) ··· ρ(Kn,u) lim · ε→0+ ε ρε

−1 ρ(K1,u) 1 − φ φ(1) − εφ ρ ρ(K1,u) ε = ρ(K2,u) ··· ρ(Kn,u) lim ρε · φ · . ε→0+ ρε ρ(K ,u) φ(1) − φ(1) − εφ 1 ρε Noting that y → 1+ as ε → 0+, we have

(ρ − ρ(L ,u))ρ(K ,u) ··· ρ(K ,u) lim ε 1 2 n ε→0+ ε

11 ρ(K1,u) 1 − y = ρ(L1,u)ρ(K2,u) ··· ρ(Kn,u)φ lim ρ(L ,u) y→1+ φ(1) − φ(y) 1 1 ρ(K ,u) = φ 1 ρ(L ,u)ρ(K ,u) ··· ρ(K ,u), (4.3) φ′ (1) ρ(L ,u) 1 2 n r 1 where − ρ(K ,u) y = φ 1 φ(1) − εφ 1 . ρ ε The equation (4.2) follows immediately from (2.3) with (4.3). This theorem plays a central role in deriving the Orlicz multiple dual mixed volume. Hence, we give the second proof. Proof Since

dρε d ρ(L1,u) =   dε dε − ρ(K ,u) φ 1 φ(1) − εφ 1    ρε    − dφ 1(y) ρ(K ,u) dφ(z) ρ(K ,u) dρ ρ(L ,u) φ 1 − ε · 1 ε 1 ρε 2 dy dz ρε dε = 2 . − ρ(K ,u) φ 1 φ(1) − εφ 1 ρ ε where ρ(K ,u) y = φ(1) − εφ 1 , ρ ε and ρ(K ,u) z = 1 . ρε

Hence − dφ 1(y) ρ(K ,u) ρ(L ,u) φ 1 dρ 1 dy ρε ε = . (4.4) 2 −1 dε − ρ(K ,u) ρ(K ,u)ρ(L ,u) dφ (y) dφ(z) φ 1 φ(1) − εφ 1 + ε · 1 1 ρ ρ2 dy dz ε ε Since (ρ − ρ(L ,u))ρ(K ,u) ··· ρ(K ,u) lim ε 1 2 n ε→0+ ε

ρε − ρ(L1,u) = ρ(K2,u) ··· ρ(Kn,u) lim ε→0+ ε

dρε = ρ(K2,u) ··· ρ(Kn,u) lim . (4.5) ε→0+ dε On the other hand − dφ 1(y) φ−1(φ(1) + △y) − 1 lim = lim ε→0+ dy △y→0+ △y ω − 1 = lim ω→1+ φ(ω) − φ(1)

12 1 = ′ , (4.6) φr(1) where ω = φ−1(φ(1) + △y). From (4.4), (4.5), (4.6) and Lemma 4.2, we obtain

(ρε − ρ(L1,u))ρ(K2,u) ··· ρ(Kn,u) 1 ρ(K1,u) lim = ′ · φ ρ(L1,u)ρ(K2,u) ··· ρ(Kn,u). (4.7) ε→0+ ε φ (1) ρ(L ,u) r 1 From (2.3) and (4.7), the equation (4.2) follows easy. n For any L1,K1, ··· ,Kn ∈ S and φ ∈ C, the integral on the right-hand side of (4.2) denoting by

Vφ(L1,K1, ··· ,Kn), and hence this new Orlicz multiple dual mixed volume Vφ(L1,K1, ··· ,Kn) has been born. e e n Lemma 4.4 If L1,K1, ··· ,Kn ∈ S and φ ∈C, then

′ d Vφ(L1,K1, ··· ,Kn)= φr(1) · V (L1+φε · K1,K2, ··· ,Kn). (4.8) dε + ε=0

e e b Proof This yields immediately from the Deﬁnition 4.1 and the variational formula of volume (4.2) Lemma 4.5 Let K,L ∈ Sn and φ ∈C, then

V (K,K+ ε · L) − V (K) 1 V (K+ ε · L) − V (K) lim 1 φ = lim φ . (4.9) ε→0+ ε n ε→0+ ε e b − b Proof Suppose ε> 0, K,L ∈ Sn and u ∈ Sn 1, let

ρ¯ε = ρ(K+φε · L,u).

From (1.3), (2.1), (2.2) and (4.4), we obtain b

V (K,K+ ε · L) − V (K) 1 ρ(K+ ε · L),u)ρ(K,u)n−1 − ρ(L,u)n lim 1 φ = lim φ dS(u) ε→0+ ε n n−1 ε→0+ ε ZS e b b 1 − dρ¯ = ρ(K,u)n 1 lim ε dS(u) n n−1 ε→0+ dε ZS 1 ρ(L,u) n = ′ φ ρ(K,u) dS(u) nφ (1) n−1 ρ(K,u) r ZS 1 V (K+ ε · L) − V (K) = lim φ . n ε→0+ ε n b Lemma 4.6 Let L1,K1,...,Kn ∈ S and φ ∈C, then

Vφ(L1,K1, ··· ,Kn)= Vφ(K,L), (4.10)

e e if K2 = ··· = Kn = K, L1 = K and K1 = L.

13 Proof On the one hand, putting K2 = ··· = Kn = K, L1 = K and K1 = L in (4.8), and noting Lemma 4.5 and (1.3), it follows that

′ d Vφ(L1,K1, ··· ,Kn)= φr(1) V (L1+φε · K1,K2, ··· ,Kn) dε + ε=0

e e b ′ V1(K,K+φε · L) − V (K) = φr(1) lim ε→0+ ε ′ e φ (1) V (K+ εb· L) − V (K) = r lim φ n ε→0+ ε b = Vφ(K,L). (4.11)

On the other hand, let K2 = ··· =e Kn = K, L1 = K and K1 = L, from Deﬁnition 4.1 and (1.3), then

1 ρ(K1,u) Vφ(L1,K1, ··· ,Kn)= φ ρ(L1,u)ρ(K2,u) ··· ρ(Kn,u)dS(u) n n−1 ρ(L ,u) ZS 1 e 1 ρ(L,u) = φ ρ(K,u)ndS(u) n n−1 ρ(K,u) ZS = Vφ(K,L). (4.12)

Combining (4.11) and (4.12), thise shows that

Vφ(L1,K1, ··· ,Kn)= Vφ(K,L),

e e if K2 = ··· = Kn = K, L1 = K and K1 = L. n Lemma 4.7 [60] If Ki,Li ∈ S and Ki → K, Li → L as i → ∞, then

a · Ki+φb · Li → a · K+φb · L, as i → ∞, (4.13)

for all a and b. b b n Lemma 4.8 If L1,K1,...,Kn,K,L ∈ S , λ1, ··· , λn ≥ 0 and φ ∈C, then

(1) Vφ(L1,K1, ··· ,Kn) > 0 (2) V (K ,K ,K , ··· ,K )= φ(1)V (K , ··· ,K ) eφ 1 1 2 n 1 n (3) V (K,K, ··· ,K)= φ(1)V (K) eφ e (4) e Vφ(λ1L1, λ1K1, λ2K2 ··· , λnKn)= λ1 ··· λnVφ(L1,K1, ··· ,Kn).

(5) e e

Vφ(L1,K1, λ1K+λ2L,K3 ··· ,Kn)

e = λ1eVφ(L1,K1,K,K3, ··· ,Kn)+ λ2Vφ(L1,K1,L,K3, ··· ,Kn).

14 This shows the Orlicz multiple mixed volume Vφ(K1, ··· ,Kn,Ln) is linear in its back (n−1) variables.

(6) Vφ(L1,K1, ··· ,Kn) is continuous. Proof From Deﬁnition 4.1, it immediately gives (1), (2), (3) and (4). e From Deﬁnition 4.1, combining the following fact

ρ(λ1K+λ2L, ·)= λ1ρ(K, ·)+ λ2ρ(L, ·),

it yields (5) directly. e

Suppose Li1 → L1, Kij → Kj as i → ∞ where j =1,...,n, combining Deﬁnition 4.1 and Lemma 4.6 with the following facts

V (Li1+φε · Ki1,Ki2, ··· ,Kin) → V (L1+φε · K1,K2, ··· ,Kn) and e b e b

V (Li1,Ki2, ··· ,Kin) → V (L1,K2, ··· ,Kn) as i → ∞, it yield (6) directly. e e Lemma 4.9 [60] Suppose K,L ∈ Sn and ε> 0. If φ ∈C, then for A ∈ GL(n)

A(K+φε · L)= AK+φε · AL. (4.14)

We easy ﬁnd that Orlicz multiple duale mixed volumeeVφ(L1,K1,K2, ··· ,Kn) is invariant under si- multaneous unimodular centro-aﬃne transformation. e n Lemma 4.10 If L1,K1,...,Kn ∈ S and φ ∈C, then for A ∈ SL(n),

Vφ(AL1, AK1, ··· , AKn)= Vφ(L1,K1, ··· ,Kn). (4.15)

Proof From (4.8) and Lemmae 4.8, we have, for A ∈eSL(n),

Vφ(AL1, AK1, ··· , AKn)

e ′ V (AL1+φε · AK1, AK2, ··· , AKn) − V (AL1, AK2, ··· , AKn) = φr(1) lim ε→0+ ε e b e ′ V (A(L1+φε · K1), AK2, ··· , AKn) − V (L1,K2, ··· ,Kn) = φr(1) lim ε→0+ ε e b e ′ V (L1+φε · K1,K2, ··· ,Kn) − V (L1,K2, ··· ,Kn) = φr(1) lim ε→0+ ε e b e = Vφ(L1,K1, ··· ,Kn).

For the conveniencee of writing, when K1 = ··· = Ki = K, Ki+1 = ··· = Kn = L, Ln = M, the Orlicz multiple dual mixed volume Vφ(K, ··· ,K,L, ··· ,L,M), with i copies of K, n − i copies of L and 1 copy of M, will be denoted by V (K [i],L [n − i],M). φ e e 15 Lemma 4.11 If K,L ∈ Sn and φ ∈C, and 0 ≤ i

1 ρ(L,u) n−i Vφ(K,L,K [n − i − 1],B [i]) = φ ρ(K,u) dS(u). (4.16) n n−1 ρ(K,u) ZS Proof On thee one hand, putting L1 = K, K1 = L,K2 = ··· = Kn−i = K and Kn−i+1 = ··· = Kn =

B in (4.8), from (2.4), (2.5), (4.4) and (4.6), we obtain for ϕ1, ϕ2 ∈ Φ

′ Wi(K,K+φε · L) − Wi(K) Vφ(K,L,K [n − i − 1],B [i]) = φr(1) lim ε→0+ ε f b n−i−1f n−i e 1 ′ ρ(K+φε · L)ρ(K,u) − ρ(K,u) = φr(1) lim dS(u) n n−1 ε→0+ ε ZS ′ b 1 n−i−1 dρε = φl(1) ρ(K,u) lim dS(u) n n−1 ε→0+ ε ZS 1 ρ(L,u) − = φ ρ(K,u)n idS(u). (4.17) n n−1 ρ(K,u) ZS On the one hand, putting L1 = K, K1 = L,K2 = ··· = Kn−i = K and Kn−i+1 = ··· = Kn = B in Deﬁnition 4.1, we have

1 ρ(L,u) n−i Vφ(K,L,K [n − i − 1],B [i]) = φ ρ(K,u) dS(u). (4.18) n n−1 ρ(K,u) ZS Combining (4.17)e and (4.18), (4.16) yields easy.

Here, we denote the Orlicz multiple dual mixed volume Vφ(K,L,K [n − i − 1],B [i]) by Wφ,i(K,L), and call W (K,L) as Orlicz dual quermassintegral of star bodies K and L. When i = 0, Orlicz dual φ,i e f quermassintegral W (K,L) becomes Orlicz dual mixed volume V (K,L). f φ,i φ Remark 4.12 When φ(t)= t−p, p = 1 and L = K , from the integral representation for Orlicz mul- f 1 1 e tiple dual mixed volume Vφ(L1,K1, ··· ,Kn) (see Deﬁnition 4.1), it follows easy that Vφ(L1,K1, ··· ,Kn)= V (K , ··· ,K ). On the other hand, when φ(t) = t−p, p = 1 and L = K , from (4.8) and noting that 1 n e 1 1 e φ′ (1) = −1, hence er

V (K1, ··· ,Kn) − V (K1+ε · K1,K2, ··· ,Kn) V (K1, ··· ,Kn) = lim . (4.19) ε→0+ ε e e b This is very interestinge for the usually dual mixed volume of this form. −p Remark 4.13 When φ(t)= t , p ≥ 1, write the Orlicz multiple dual mixed volume Vφ(L1,K1, ··· ,Kn)

as V− (L ,K , ··· ,K ) and call as the L -multiple dual mixed volume, from Deﬁnition 4.1, it easy yields p 1 1 n p e 1 −p 1+p e V−p(L1,K1, ··· ,Kn)= ρ(K1,u) ρ(L1,u) ρ(K2,u) ··· ρ(Kn,u)dS(u). n n−1 ZS −p When ϕ(t)e = t and p ≥ 1, from (4.8), we get the following expression of Lp-multiple dual mixed volume.

1 V (L1+pε · K1,K2, ··· ,Kn) − V (L1,K2, ··· ,Kn) V−p(L1,K1, ··· ,Kn) = lim . −p ε→0+ ε e b e e 16 Lemma 4.14 (Jensen’s inequality) Let µ be a probability measure on a space X and g : X → I ⊂ R is a µ-integrable function, where I is a possibly inﬁnite interval. If ψ : I → R is a convex function, then

ψ(g(x))dµ(x) ≥ ψ g(x)dµ(x) . (4.20) ZX ZX If ψ is strictly convex, equality holds if and only if g(x) is constant for µ-almost all x ∈ X (see [24, p.165]).

5 Dual Orlicz-Aleksandrov-Fenchel inequality

n Theorem 5.1 If L1,K1, ··· ,Kn ∈ S and φ ∈C, then

V (K1, ··· ,Kn) Vφ(L1,K1, ··· ,Kn) ≥ V (L1,K2, ··· ,Kn)φ . (5.1) V (L1,K2, ··· ,Kn)! e e e If φ is strictly convex, equality holds if and only if K1 and L1 aree dilates. n n−1 Proof For K1, ··· ,Kn ∈ S and any u ∈ S , since

1 ρ(K1,u) ··· ρ(Kn,u)dS(u)=1, n−1 nV (K1, ··· ,Kn) ZS

ρ(K ,u) ··· ρ(K ,u) − so 1 n S(ue) is a probability measure on Sn 1. From Deﬁnition 4.1, Jensen’s inequality nV (K1, ··· ,Kn) (4.20) and (2.3), it follows that e V (L ,K , ··· ,K ) 1 φ 1 1 n = V (L1,K2, ··· ,Kn) nV (L1,K2, ··· ,Kn) e e e ρ(K1,u) × φ ρ(L1,u)ρ(K2,u) ··· ρ(Kn,u)dS(u) n−1 ρ(L ,u) ZS 1 1 ≥ φ ρ(K1,u) ··· ρ(Kn,u)dS(u) n−1 nV (L1,K2, ··· ,Kn) ZS ! V (K , ··· ,K ) = φ e 1 n . (5.2) V (L1,K2, ··· ,Kn)! e Next, we discuss the equality condition of (5.2). Suppose the equality hold in (5.2), form the equality e condition of Jensen’s inequality, it follows that if φ is strictly convex the equality in (5.2) holds if and

only if K1 and L1 are dilates. n Theorem 5.2 (Dual Orlicz-Aleksandrov-Fenchel inequality) If L1,K1, ··· ,Kn ∈ S , 1 ≤ r ≤ n and φ ∈C, then

r 1 r i=1 V (Ki ...,Ki,Kr+1,...,Kn) Vφ(L1,K1,K2, ··· ,Kn) ≥ V (L1,K2, ··· ,Kn)φ . (5.3) V (L1,K2 ...,Kn) ! Q e e e e 17 If φ is strictly convex, equality holds if and only if L1,K1,...,Kr are all dilations of each other. Proof This follows immediately from Theorem 5.1 with the dual Aleksandrov-Fenchel inequality.

−p Obviously, putting φ(t) = t , p = 1 and L1 = K1 in (5.3), (5.3) becomes the Lutwak’s dual Aleksandrov-Fenchel inequality (1.11) stated in the introduction. n Corollary 5.3 If L1,K1, ··· ,Kn ∈ S and φ ∈C, then

1 n V (K1) ··· V (Kn) Vφ(L1,K1,K2, ··· ,Kn) ≥ V (L1,K2, ··· ,Kn)φ . (5.4) n  V (L1,K2,...,Kn) !  e e   If φ is strictly convex, equality holds if and only if L1,K1,...,Ken are all dilations of each other. Proof This follows immediately from Theorem 5.2 with r = n. Corollary 5.4 If K,L ∈ Sn, 0 ≤ i

1/(n−i) Wi(L) Wφ,i(K,L) ≥ Wi(K)φ . (5.5)  Wi(K)!  f f f   If φ is strictly convex, equality holds if and only if K andfL are dilates.

Proof This follows immediately from Theorem 5.2 with r = n − i, L1 = K, K1 = L, K2 = ··· =

Kn−i = K and Kn−i+1 = ··· = Kn = B. The following inequality follows immediately from (5.5) with φ(t) = t−p and p ≥ 1. If K,L ∈ Sn, 0 ≤ i

n n+p −p V−p(K,L) ≥ V (K) V (L) , (5.7) with equality if and only if K and Leare dilates. Theorem 5.5 (Orlicz dual isoperimetric inequality) If K ∈ Sn and φ ∈C, and 0 ≤ i

1/(n−i) V (K,B,K [n − i − 1],B [i]) V (B) φ ≥ φ . (5.8) Wi(K)  Wi(K)!  e   If φ is strictly convex, equality holdsf if and only if K is a ball.f

Proof This follows immediately from (5.3) with with r = n−i, L1 = K, K1 = B, K2 ··· = Kn−i = K,

and Kn−i+1 = ··· = Kn = B.

18 −p When φ(t) = t , p ≥ 1, the Orlicz isoperimetric inequality (5.8) becomes the following Lp-dual isoperimetric inequality. If K is a star body, p ≥ 1 and 0 ≤ i

n−i n−i+p nV− (K,B) W (K) p ≥ i , (5.9) ωn ! κn ! e f with equality if and only if K is ball, and where κn denotes volume of the unit ball B, and its surface area by ωn. Putting p = 1 and i = 0 in (5.9), (5.9) becomes the following dual isoperimetric inequality. If K is a star body, then n n+1 nV− (K,B) V (K) 1 ≥ , ωn ! κn e with equality if and only if K is ball. n Theorem 5.6 If L1,K1, ··· ,Kn ∈M⊂S , and φ ∈C be strictly convex, and if either

Vφ(Q,K1,...,Kn)= Vφ(Q,L1,K2,...,Kn), forall Q ∈M, (5.10)

or e e V (K ,Q,K ,...,K ) V (L ,Q,K ,...,K ) φ 1 2 n = φ 1 2 n , forall Q ∈M, (5.11) V (K1,...,Kn) V (L1,K2,...,Kn) e e then K1 = L1. e e Proof Suppose (5.10) hold. Taking K1 for Q, then from Deﬁnition 4.1 and Theorem 5.1, we obtain

V (L1,K2,...,Kn) φ(1)V (K1,...,Kn)= Vφ(K1,L1,K2,...,Kn) ≥ V (K1,...,Kn)φ , V (K1,...,Kn) ! e e e e with equality if and only if K1 and L1 are dilates. Hence e V (L ,K ,...,K ) φ(1) ≥ φ 1 2 n , V (K1,...,Kn) ! e with equality if and only if K1 and L1 are dilates.e Since ϕ is decreasing function on (0, ∞), this follows that

V (K1,...,Kn) ≤ V (L1,K2,...,Kn),

with equality if and only if K1 ande L1 are dilates.e On the other hand, if taking L1 for Q, we similar

get V (K1,...,Kn) ≥ V (L1,K2,...,Kn), with equality if and only if K1 and L1 are dilates. Hence V (K ,...,K ) = V (L ,K ,...,K ), and K and L are dilates, it follows that K and L must be e1 n 1e 2 n 1 1 1 1 equal. e e Suppose (5.11) hold. Taking K1 for Q, then from Deﬁnition 4.1 and Theorem 5.1, we obtain

V (L ,K ,...,K ) V (K ,...,K ) φ(1) = φ 1 1 n ≥ φ 1 n , V (L1,K2,...,Kn) V (L1,K2 ...,Kn) ! e e e 19 e with equality if and only if K1 and L1 are dilates. Since ϕ is increasing function on (0, ∞), this follows that

V (L1,K2,...,Kn) ≤ V (K1,...,Kn), with equality if and only if K1 ande L1 are dilates. One the other hand, if taking L1 for Q, we similar get V (L1,K2,...,Kn) ≥ V (K1,...,Kn), with equality if and only if K1 and L1 are dilates. Hence V (L ,K ,...,K ) = V (K ,...,K ), and K and L are dilates, it follows that K and L must be 1e 2 n e1 n 1 1 1 1 equal. e e Corollary 5.7 Let K,L ∈M⊂Sn, 0 ≤ i

Wφ,i(Q,K)= Wφ,i(Q,L), for all Q ∈M, or f f W (K,Q) W (L,Q) φ,i = φ,i , for all Q ∈M, Wi(K) Wi(L) f f then K = L. f f Proof This yields immediately from Theorem 5.6 and Lemma 4.11. Remark 5.8 When φ(t) = t−p and p = 1, the Orlicz dual Aleksandrov-Fenchel inequality (5.3) n becomes the following inequality. If L1,K1, ··· ,Kn ∈ S and 1 ≤ r ≤ n, then

2 V (L1,K2, ··· ,Kn) V−1(L1,K1,K2, ··· ,Kn) ≥ r , (5.12) 1 V (Kei ...,Ki,Kr+1,...,Kn) r e i=1 Y e with equality if and only if L1,K1,...,Kr are all dilations of each other.

Putting L1 = K1 in (5.12) and noting that V−1(K1,K1,K2, ··· ,Kn) = V (K1,K2, ··· ,Kn), (5.12) becomes the dual Aleksandrov-Fenchel inequality (1.11). Putting r = n in (5.12), (5.12) becomes the e e following inequality.

n 2n −1 V−1(L1,K1, ··· ,Kn) ≥ V (L1,K2 ··· ,Kn) (V (K1) ··· V (Kn)) , (5.13)

e e with equality if and only if L1,K1,...,Kn are all dilations of each other. Putting L1 = K, K1 = L and n K2 = ··· = Kn = K in (5.13), (5.13) becomes the well-known Minkowski inequality. If K,L ∈ S , then

n n+1 −1 V−1(K,L) ≥ V (K) V (L) , (5.14) with equality if and only if K and Leare dilates. Obviously, inequality (5.12) in special case yields also n the following result. If K1,...,Kn ∈ S and 0 ≤ i

n−i n−i+1 −1 W−1,i(K,L) ≥ Wi(K) Wi(L) , (5.15)

f f 20 with equality if and only if K and L are dilates. When i = 0, (5.15) becomes (5.14). On the other hand, n putting L1 = K1 in (5.13), (5.13) becomes the well-known inequality. If K1,...,Kn ∈ S , then

n V (K1, ··· ,Kn) ≤ V (K1) ··· V (Kn), with equality if and only if K1,...,Ke n are all dilations of each other.

6 Dual Orlicz-Brunn-Minkowski inequality for the Orlicz harmonic addition

n Lemma 6.1 If L1,K1, ··· ,Kn ∈ S and φ ∈C, then

φ(1)V (K1+φL1,K2, ··· ,Kn)= Vφ(K1+φL1,K1, ··· ,Kn)+ Vφ(K1+φL1,L1,K2, ··· ,Kn). (6.1)

n n−1 Proofe Supposeb ε> 0, K1,L1 ∈ Se andbu ∈ S , let e b

Q = K1+φε · L1.

From Deﬁnition 4.1, (2.3) and (3.4), we have b

φ(1)V (Q,K2, ··· ,Kn)

1 ρ(K1,u) ρ(L1,u) =e φ + φ ρ(Q,u)ρ(K2,u) ··· ρ(Kn,u)dS(u) n n−1 ρ(Q,u) ρ(Q,u) ZS = Vφ(Q,K1, ··· ,Kn)+ Vφ(Q,L1,K2, ··· ,Kn). (6.2) Putting Q = K1+φLe 1 in (6.2), (6.2) changese (6.1).

Theorem 6.2 (Dual Orlicz-Brunn-Minkowski inequality for the Orlicz harmonic addition) If L1,K1, ··· ,Kn ∈ b Sn and φ ∈C, then for ε> 0

V (K ,...,K ) V (L ,K , ··· ,K ) φ(1) ≥ φ 1 n + ε · φ 1 2 n , (6.3) V (K1+φε · L1,K2, ··· ,Kn) ! V (K1+φε · L1,K2, ··· ,Kn) ! e e If φ is strictly convex, equality holds if and only if K and L are dilates. e b 1 1 e b Proof From Theorem 5.1 and Lemma 6.1, we have

φ(1)V (K1+φε · L1,K2, ··· ,Kn)

=eVφ(Kb1+φε·L1,K1, ··· ,Kn)+ε·Vφ(K1+φε·L1,L1,K2, ··· ,Kn)

V (K1,...,Kn) V (L1,K2, ··· ,Kn) ≥eV (K1b+φε·L1,K2, ··· ,Kn) φ e b + φ . ( V (K1+φε · L1,K2, ··· ,Kn) ! V (K1+φε · L1,K2, ··· ,Kn)!) e e From the equalitye b condition of Theorem 5.1, the equality in (6.3) holds if and only if K , L and K + L e b e b 1 1 1 φ 1 are dilates if φ is strictly convex, it follows that if φ is strictly convex, the equality in (6.3) holds if and b only if K1 and L1 are dilates.

21 n Theorem 6.3 (Dual Aleksandrov-Fenchel type inequality) If L1,K1, ··· ,Kn ∈ S , 0 ≤ i,j < n, 1 < r ≤ n and φ ∈C, then

r 1 r 1 V (K ,...,K ,K ,...,K ) r M(r) · V (Kj ,...,Kj,Kr+1,...,Kn) r φ(1) ≥ φ i=1 i i r+1 n + φ j=2 , Q V (K1+φL1,K2, ··· ,Kn) ! Q V (K1+φL1,K2, ··· ,Kn) ! e e (6.4) e b 1 e b where M(r)= V (L1,...,L1,Kr+1,...,Kn) r . If φ is strictly convex, equality holds if and only if L1,K1,...,Kr are all dilations of each other. e Proof This follows immediately from Theorem 6.2 and the dual Aleksandrov-Fenchel inequality.

n Corollary 6.4 (Lp-dual Brunn-Minkowski inequality) If L1,K1, ··· ,Kn ∈ S , 0 ≤ i,j

r − −p p V (K1+pL1,K2, ··· ,Kn) ≥ V (Ki,...,Ki,Kr+1,...,Kn) r i=1 Y e b e r − −p p + M(r) · V (Kj ,...,Kj,Kr+1,...,Kn) r , (6.5) j=1 Y e with equality if and only if L1,K1,...,Kr are all dilations of each other. Proof This follows immediately from (6.4) with φ(t)= t−p and p ≥ 1. Corollary 6.5 If K,L ∈ Sn, φ ∈C and 0 ≤ i

1 1 − − W (K) n i W (L) n i φ(1) ≥ φ i + φ i . (6.6)  Wi(K+φL)!   Wi(K+φL)!  f f     If φ is strictly convex, equality holdsf if andb only if K and L aref dilates.b

Proof This follows immediately from Theorem 6.3 with r = n − i, K2 = ··· = Kn−i = K+φL

Kn−i+1 = ··· = Kn = B. b The following inequality follows immediately from (6.6) with φ(t) = t−p and p ≥ 1. If K,L ∈ Sn, 0 ≤ i

−p/(n−i) −p/(n−i) −p/(n−i) Wi(K+pL) ≥ Wi(K) + Wi(L) , with equality if and onlyf if K andb L are dilates.f f n Corollary 6.6 If L1,K1, ··· ,Kn ∈ S and φ ∈C, then

1 1 V (K ) ··· V (K ) n V (L )V (K ) ··· V (K ) n φ(1) ≥ φ 1 n + φ 1 2 n . (6.7) n n  V (K1+φL1,K2, ··· ,Kn) !   V (K1+φL1,K2, ··· ,Kn) !      If φ is strictly convex,e equalityb holds if and only if L1,K1,...,Ke n bare all dilations of each other.

22 Proof This follows immediately from Theorem 6.3 with r = n. n Corollary 6.7 If L1,K1, ··· ,Kn ∈ S and p ≥ 1, then

− − −p p p V (K1+pL1,K2 ··· Kn) ≥ (V (K1) ··· V (Kn)) n + (V (L1)V (K2) ··· V (Kn)) n , (6.8)

e with equality if andb only if L1,K1,...,Kn are all dilations of each other. Proof This follows immediately from (6.7) with φ(t)= t−p and p ≥ 1.

Putting K2 = ··· = Kn = K1+pL1 in (6.8), (6.8) becomes Lutwak’s Lp-dual Brunn-Minkowski inequality b −p/n −p/n −p/n V (K+pL) ≥ V (K) + V (L) ,

with equality if and only if K and Lbare dilates. n Corollary 6.8 If L1,K1, ··· ,Kn ∈ S , 1 ≤ r ≤ n and φ ∈C, then

r 1 r i=1 V (Ki ...,Ki,Kr+1,...,Kn) Vφ(L1,K1,K2, ··· ,Kn) ≥ V (L1,K2, ··· ,Kn)φ . (6.8) V (L1,K2 ...,Kn) ! Q e e e If φ is strictly convex, equality holds if and only if L1,K1,...,Ker are all dilations of each other. Proof Let

Kε = L1+φε · K1.

From (4.8), dual Orlicz-Brunn-Minkowski inequalityb (6.3) and dual Aleksandrov-Fenchel inequality, we obtain

1 d ′ · Vφ(L1,K1,...,Kn)= V (Kε,K2, ··· ,Kn) φ (1) dε + + ε=0

e e V (K ,K , ··· ,K ) − V (L ,K , ··· ,K ) = lim ε 2 n 1 2 n ε→0+ ε e e V (L ,K , ··· ,K ) V (L1,K2, ··· ,Kn) 1 − 1 2 n φ(1) − φ V (Kε,K2, ··· ,Kn) V (Kε,K2, ··· ,Kn)! = lim · e · V (Kε,K2, ··· ,Kn) ε→0+ e V (L1,K2, ··· ,Kn) ε φ(1) − φ e e V (Kε,K2, ··· ,Kn)! e e V (L ,K , ··· ,K ) e φ(1) − φ 1 2 n 1 − t V (Kε,K2, ··· ,Kn) ! = lim · lim e · lim V (Kε,K2, ··· ,Kn) t→0+ φ(1) − (t) ε→0+ ε ε→0+ e 1 V (K1,K2, ··· ,Kn) e ≥ ′ · lim φ · V (L1,K2, ··· ,Kn) → + φ+(1) ε 0 V (Kε,K2, ··· ,Kn) ! e e 1 V (K1,Ke 2, ··· ,Kn) = ′ · φ · V (L1,K2, ··· ,Kn) φ+(1) V (L1,K2, ··· ,Kn) ! e e e 23 r 1 r 1 i=1 V (Ki ...,Ki,Kr+1,...,Kn) ≥ ′ · φ · V (L1,K2, ··· ,Kn). (6.9) φ+(1) V (L1,K2 ...,Kn) ! Q e From (6.9), inequality (5.1) easy follows. From the equality conditionse of the dual Orlicz-Brunn-Minkowski e inequality (6.3) and dual Aleksandrov-Fenchel inequality, it follows that if φ is strictly convex, the equality in (6.9) holds if and only if L1,K1,...,Kn are all dilations of each other. This proof is complete.

Availability of data and material All data generated or analysed during this study are included in this published article. Competing interests The author declare that he has no competing interests. Funding The author’s research is supported by the Natural Science Foundation of China (11371334, 10971205). Authors’s contributions C-JZ contributed to the main results. The author read and approved the ﬁnal manuscript. Acknowledgements The ﬁrst author expresses his gratitude to Professors G. Leng and W. Li for their valuable helps.

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