Orlicz mixed quermassintegrals

Mathematics

.

.

doi: 10.1007/s11425-014-4812-4

c Science China Press and Springer-Verlag Berlin Heidelberg 2014 math.scichina.com link.springer.com

Orlicz mixed quermassintegrals

Dedicated to Professor Ren De-lin on the Occasion of his80th Birthday

XIONG Ge

& ZOU Du

Department of Mathematics, Shanghai University, Shanghai200444, China Email: [email protected], [email protected]

Received September 26, 2013; accepted January 8, 2014; published online April 4, 2014

Abstract The notion of mixed quermassintegrals in the classical Brunn-Minkowski theory is extended to that of Orlicz mixed quermassintegrals in the Orlicz Brunn-Minkowski theory. The analogs of the classical Cauchy- Kubota formula, the Minkowski isoperimetric inequality and the Brunn-Minkowski inequality are established for this new Orlicz mixed quermassintegrals.

Keywords Orlicz Brunn-Minkowski theory, quermassintegral, Minkowski’s isoperimetric inequality, integral geometry

MSC(2010) 52A39, 52A40, 52A22

Citation: Xiong G, Zou D. Orlicz mixed quermassintegrals. Sci China Math, 2014, 57: 2549–2562, doi: 10.1007/

s11425-014-4812-4

1 Introduction

Beginning with the groundbreaking articles [11, 20, 21] and the very recent work [9], the classical Brunn- Minkowski theory of convex bodies (see, e.g., [8, 10, 28, 30]) was extended to the Orlicz stage, which is known as the Orlicz Brunn-Minkowski theory. Analogous to the way that Orlicz spaces generalize L_p spaces (see [23]), it represents a generalization of theL_p Brunn-Minkowski theory, which emerged in the early 1960s (see [6]), began largely with the initial works [14, 15] in the mid 1990s, and expanded rapidly thereafter (see, e.g., [2–4, 7, 12, 13, 16–19, 22, 25, 27, 29, 31–34]).

A convex body inRⁿ, the standard Euclideann-space, is a compact convex set with non-empty interior.

Associated with a convex bodyQinRⁿare the quermassintegrals W0(Q), W1(Q), . . . , W_n(Q), which are defined by letting W0(Q) = vol_n(Q), the volume of Q; W_n(Q) = ω_n =π^n/2/Γ(1 +n/2), the volume of the unit ballB in Rⁿ; and forj= 1,2, . . . , n−1 by

W_n−j(Q) = ω_n ω_j

Gn,j

vol_j(Q|ξ)dσ_n,j(ξ), (1.1)

where G_n,j, σ_n,j, vol_j and Q| ξ denote the Grassmannian manifold of j-dimensional linear subspaces of Rⁿ, the normalized Haar measure onG_n,j, the j-dimensional volume and the orthogonal projection ofQontoξ, respectively. Equation (1.1) is the well-known Cauchy-Kubota formula.

That studying the first variation of quermassintegrals is an effective approach to find new geometric quantities or measures induced by convex bodies. In the late 1930s, Aleksandrov [1] and Fenchel and

∗Corresponding author

Jessen [5] independently proved that for a convex body Qin Rⁿ andj = 0,1, . . . , n−1, there exists a regular Borel measureS_n−j−1(Q,·) onSⁿ⁻¹, the unit sphere inRⁿ, such that for any convex bodyP,

W_j(Q, P) := 1 n−j

d dε

ε=0⁺

W_j(Q+εP) = 1 n

Sⁿ⁻¹h_P(u)dS_n−1−j(Q, u),

whereh_P is the support function ofP, andQ+εP ={x+εy:x∈Q, y∈P}.The quantityW_j(Q, P) is called thej-thmixed quermassintegral ofQand P.

LetK, Lbe convex bodies inRⁿ with the origin in their interiors, andε0. Recall thatK+_pε·_pL, theL_p combination ofK andL, is defined by

h_K+_pε·pL= (h_K^p+εh_L^p)^1p.

In [14], forj= 0,1, . . . , n−1, Lutwak considered theL_pfirst variation of thej-th quermassintegral W_p,j(K, L) := p

n−j d dε

ε=0⁺

W_j(K+_pε·_pL).

The quantityW_p,j(K, L) is called theL_p mixed quermassintegral ofK andL. In particular,W_p,0(K, L) is also denoted byV_p(K, L), theL_p mixed volume ofK andL.

The aim of this paper is to extend the notion of mixed quermassintegrals to the Orlicz setting. The above cited works [11, 20, 21], and especially the work [9], make it apparent that the time is ripe.

Throughout this paper, we consider a Young function ϕ : [0,∞)→ [0,∞), i.e., ϕ is convex, strictly increasing, withϕ(0) = 0. Letα, β >0. It was shown by Gardner, Weil and Hug [9] that the function h:Rⁿ→[0,∞) given by

h(x) = inf

λ >0 :αϕ

h_K(x) λ

+βϕ

h_L(x) λ

ϕ(1)

, forx∈Rⁿ

is positive definite, homogeneous of order 1 and subadditive, and therefore is precisely the support function of a unique convex body,α·ϕK+_ϕβ·ϕL, with the origin in its interior. The convex bodyα·ϕK+_ϕβ·ϕL is called theOrlicz combination ofKandL. Ifϕ(t) =t^p,p1, then the Orlicz combination reduces to theL_p combination and particularly the classical Minkowski combination for p= 1. For brevity, when α= 1 andα=β= 1, we writeα·ϕK+_ϕβ·ϕLasK+_ϕβ·ϕLandK+_ϕL, respectively.

In Section 3, we compute the Orlicz first variations of quermassintegrals W_ϕ,j(K, L) := ϕ₋(1)

n−j d dε

ε=0⁺

W_j(K+_ϕε·_ϕL), j= 0,1, . . . , n−1, and obtain their integral representation.

Theorem 1. SupposeKandLare convex bodies inRⁿ containing the origin in their interiors, and ϕ is a Young function. Then for each j= 0,1, . . . , n−1,

d dε

ε=0⁺

W_j(K+_ϕε·ϕL) = n−j nϕ₋(1)

Sⁿ⁻¹ϕ

h_L(u) h_K(u)

h_K(u)dS_n−j−1(K, u).

Naturally, the quantityW_ϕ,j(K, L) is called thej-thOrlicz mixed quermassintegralofKandLregard- ingϕ. In particular,W_ϕ,0(K, L) is theOrlicz mixed volume V_ϕ(K, L) (see [9]) . Obviously, if ϕ(t) =t^p, p1, thenW_ϕ,j(K, L) =W_p,j(K, L).

In Section 4, from the viewpoint of integral geometry (see, e.g., [24, 26]), we show the probabilistic essence of Orlicz mixed quermassintegrals. The classical Cauchy-Kubota formula has the following natural Orlicz extension:

Theorem 2. SupposeKandLare convex bodies inRⁿ containing the origin in their interiors, and ϕ is a Young function. Then for 1qj < n,

W_ϕ,j(K, L) = ω_n ω_n−q

Gn,n−q

W_ϕ,j−q^(n−q)(K|ξ, L|ξ)dσ_n,n−q(ξ),

whereW_ϕ,j−q^(n−q)(K|ξ, L|ξ)is the Orlicz mixed quermassintegral of the(n−q)-dimensional convex bodies K|ξ andL|ξ in the subspaceξ.

In Section 5, the Minkowski isoperimetric inequality and the Brunn-Minkowski inequality for quer- massintegrals are generalized to the Orlicz setting.

Theorem 3. SupposeKandLare convex bodies inRⁿ containing the origin in their interiors, and ϕ is a Young function. Then for each j= 0,1, . . . , n−1,

W_ϕ,j(K, L) W_j(K) ϕ

W_j(L) W_j(K)

_n−j¹ ,

and

ϕ(1)ϕ

W_j(K) W_j(K+_ϕL)

_n−j¹ +ϕ

W_j(L) W_j(K+_ϕL)

_n−j¹ .

If ϕis strictly convex, each equality holds if and only ifK andLare dilates.

Ifϕ(t) =t^p,p1, then these inequalities reduce to theL_p Minkowski isoperimetric inequality W_p,j(K, L)W_j(K)^{n−j−pn−j} W_j(L)^n−jp

and theL_p Brunn-Minkowski inequality

W_j(K+_pL)^n−jp W_j(K)^n−jp +W_j(L)^n−jp , respectively.

2 Preliminaries

In order to keep the paper self-contained, we list here some basic facts about convex bodies that are needed in our investigations. The setting is the standard Euclideann-space,Rⁿ. As usual,x·y denotes the standard inner product ofxandy inRⁿ.

The support functionh_K:Rⁿ→Rof a convex bodyKinRⁿis defined byh_K(x) = max{x·y:y∈K} forx∈Rⁿ. The notation Kⁿ_o denotes the class of convex bodies inRⁿ which contain the origin in their interiors. The setKⁿ_o is often equipped with the Hausdorff metric δ_H, which is defined by

δ_H(K1, K2) = max

Sⁿ⁻¹|h_K1−h_K2|, forK1, K2∈ Kⁿ_o.

The classical Aleksandrov-Fenchel-Jessen surface area measure, S_n−1(Q,·), of a convex body Q, is defined as the unique Borel measure onSⁿ⁻¹such that

Sⁿ⁻¹f(u)dS_n−1(Q, u) =

∂Qf(ν_Q(y))dHⁿ⁻¹(y),

for each continuousf :Sⁿ⁻¹→R, whereν_Q is the unit outer normal of∂Qat x∈∂Q. It is noted that ν_Q(x) exists forHⁿ⁻¹-almost allx∈∂Q.

The n measures S0(Q,·), S1(Q,·), . . . , S_n−1(Q,·) appeared in Section 1 are the measures, which are such that for eachε >0 and for each Borel subsetω⊆Sⁿ⁻¹,

S_n−1(Q+εB, ω) =

n−1 j=0

n−1 j

S_n−1−j(Q, ω)ε^j.

The measureS_j(Q,·) is called thej-th area measureofQ. The (n−1)-th area measure is just the surface area measure. The measure S0(Q,·) is independent of the body Q, and is just the spherical Lebesgue measure onSⁿ⁻¹.

Throughout this paper, the notation ϕ and Φ denote a Young function and the class of all Young functions, respectively.

ForK, L∈ Kⁿ_o andα, β >0, theOrlicz combination,α·ϕK+_ϕβ·ϕL, is the convex body with support function

h_α·ϕK+_ϕβ·ϕL(x) = inf

λ >0 :αϕ

h_K(x) λ

+βϕ

h_L(x) λ

ϕ(1)

.

It is noted thath_α·ϕK+_ϕβ·ϕL can be defined for all nonzerox∈Rⁿ by the equation αϕ

h_K(x) h_α·ϕK+_ϕβ·ϕL(x)

+βϕ

h_L(x) h_α·ϕK+_ϕβ·ϕL(x)

=ϕ(1), and byh_α·ϕK+_ϕβ·ϕL(0) = 0. Ifϕ(t) =t^p, 1p <∞, then

α·_ϕK+_ϕβ·_ϕL=α·_pK+_pβ·_pL.

Associated with each positive continuous functionf onSⁿ⁻¹is its Aleksandrov body,A(f), defined by A(f) =

u∈Sⁿ⁻¹

{x∈Rⁿ:x·uf(u)}. Obviously,A(h_Q) =Qfor any convex bodyQin Rⁿ.

To obtain the Orlicz first variation of volume, it is crucial to use Aleksandrov’s variational principle (see [3, Lemma 3.1], [9, Lemma 8.3] or [11, Lemma 1]).

Lemma 2.1. Let I⊂Rbe an interval containing0 and some positive number and the function h_t(u) =h(t, u) :I×Sⁿ⁻¹→(0,∞)

be continuous and such that the convergence in

h₊(0, u) = lim

t→0⁺

h(t, u)−h(0, u) t is uniform on Sⁿ⁻¹. Then

d dt

t=0⁺

vol_n(A(h_t)) =

Sⁿ⁻¹h+(0, u)dS_n−1(A(h0), u).

According to [9, Lemmas 8.2 and 8.4], it gives that

K+_ϕε·ϕL→K as ε→0⁺, (2.1)

and ∂

∂ε

ε=0⁺

h_K+_ϕε·ϕL(u) = 1 ϕ₋(1)ϕ

h_L(u) h_K(u)

h_K(u), (2.2)

uniformly foru∈Sⁿ⁻¹. Note that

h_ε(u) =h_K+_ϕε·ϕL(u) : [0,∞)×Sⁿ⁻¹→(0,∞)

is continuous. Hence, from Lemma 2.1 as well as the fact thatA(h_ε) =K+_ϕε·ϕLforε0, it follows

that d

dε

ε=0+

vol_n(K+_ϕε·ϕL) = 1 ϕ₋(1)

Sⁿ⁻¹ϕ

h_L(u) h_K(u)

h_K(u)dS_n−1(K, u). (2.3) The above formula allows us to define theOrlicz mixed volume, V_ϕ(K, L), ofK, L∈ K_oⁿ regardingϕ, by

V_ϕ(K, L) = 1 n

Sⁿ⁻¹

ϕ

h_L(u) h_K(u)

h_K(u)dS_n−1(K, u).

Ifϕ(t) =t^p, 1p <∞, thenV_ϕ(K, L) =V_p(K, L),theL_p mixed volume ofK andL.

We conclude this section with the next lemma, which will be used in the proof of Theorem 3.1.

Lemma 2.2. Let K, L∈ K_oⁿ andϕ∈Φ. Then for0ε1< ε2<∞, there is the inclusion K+_ϕε1·_ϕL⊂K+_ϕε2·_ϕL.

Proof. Letu∈Sⁿ⁻¹ be arbitrary but fixed. Forε0, define the functionψ_ε: (0,∞)→(0,∞) by ψ_ε(λ) =ϕ(λ⁻¹h_K(u)) +εϕ(λ⁻¹h_L(u)).

It is easy to check thatψ_ε(λ) is continuous and strictly decreasing inλ∈(0,∞),

λ→lim0⁺ψ_ε(λ) =∞, and lim

λ→∞ψ_ε(λ) = 0.

These facts allow us to take the functionψ_ε⁻¹ : (0,∞)→(0,∞), the inverse function ofψ_ε. Note that for 0ε1< ε2<∞,

ϕ(1) =ψ_ε1(h_K+_ϕε1·ϕL(u))< ψ_ε2(h_K+_ϕε1·ϕL(u)).

Hence,

h_K+_ϕε2·ϕL(u) =ψ⁻_ε2¹(ϕ(1))> ψ_ε⁻₂¹(ψ_ε2(h_K+_ϕε1·ϕL(u))) =h_K+_ϕε1·ϕL(u).

Sinceuis arbitrary, we complete the proof.

3 Orlicz mixed quermassintegrals

Formula (2.3) expresses the Orlicz first variation of then-dimensional volume. For the completeness of the study, we consider

d dε

ε=0⁺

W_j(K+_ϕε·_ϕL), j= 0,1, . . . , n−1.

The following theorem gives their integral representations.

Theorem 3.1. SupposeK, L∈ Kⁿ_o andϕ∈Φ. Then for each j= 0,1, . . . , n−1, d

dε

ε=0⁺

W_j(K+_ϕε·ϕL) = n−j nϕ₋(1)

Sⁿ⁻¹ϕ

h_L(u) h_K(u)

h_K(u)dS_n−j−1(K, u). (3.1) Letting ϕ(t) =t and ϕ(t) =t^p with p1 in (3.1), it gives the integral representations ofW_j(K, L) andW_p,j(K, L), respectively.

To prove the theorem, Minkowski’s isoperimetric inequality forW_j(K, L) is indispensable: For convex bodiesK, Lin Rⁿ andj= 0,1, . . . , n−1,

W_j(K, L)W_j(K)^{n−j−1n−j} W_j(L)^n−j1 , (3.2) with equality if and only ifK andLare homothetic.

Proof. For brevity, we writeK+_ϕε·ϕLasK_ε, and define g: [0,∞)→(0,∞) by g(ε) =W_j(K_ε)^n−j1 .

The continuity ofW_j:Kⁿ_o →Ras well as (2.1) implies thatgis continuous at 0. By Lemma 2.2 and the definition of quermassintegrals, we haveW_j(K_ε1)< W_j(K_ε2), for 0ε1< ε2<∞. Thus, g(ε)^n−j and g(ε) are both increasing inε. Hence, we get

g(0)^n−j−1lim inf

ε→0⁺

g(ε)−g(0)

ε =W_j(K)^{n−j−1n−j} lim inf

ε→0⁺

W_j(K_ε)^n−j1 −W_j(K)^n−j1 ε

= lim inf

ε→0⁺ W_j(K_ε)^{n−j−1n−j} W_j(K_ε)^n−j1 −W_j(K)^n−j1

ε .

According to (3.2), it immediately yields lim inf

ε→0⁺

g(ε)−g(0)

ε g(0)^{−(n−j−1)}lim inf

ε→0⁺

W_j(K_ε)−W_j(K_ε, K)

ε . (3.3)

Similarly, it has

lim sup

ε→0⁺

g(ε)−g(0)

ε g(0)^{−(n−j−1)}lim sup

ε→0⁺

W_j(K, K_ε)−W_j(K)

ε . (3.4)

The weak continuity of surface area measures as well as (2.1) implies that S_n−j−1(K_ε,·) weakly con- verges toS_n−j−1(K,·) onSⁿ⁻¹, asε→0⁺. This and (2.2) yield that

ε→lim0⁺

W_j(K_ε)−W_j(K_ε, K)

ε = lim

ε→0⁺

1 n

Sⁿ⁻¹

h_Kε(u)−h_K(u)

ε dS_n−j−1(K_ε, u)

= 1 n

Sⁿ⁻¹ lim

ε→0⁺

h_Kε(u)−h_K(u)

ε dS_n−j−1(K, u)

= 1

nϕ₋(1)

Sⁿ⁻¹ϕ

h_L(u) h_K(u)

h_K(u)dS_n−j−1(K, u).

That is,

ε→lim0⁺

W_j(K_ε)−W_j(K_ε, K)

ε = 1

nϕ₋(1)

Sⁿ⁻¹ϕ

h_L(u) h_K(u)

h_K(u)dS_n−j−1(K, u). (3.5) Similarly, we have

ε→lim0⁺

W_j(K, K_ε)−W_j(K)

ε = 1

nϕ₋(1)

Sⁿ⁻¹ϕ

h_L(u) h_K(u)

h_K(u)dS_n−j−1(K, u). (3.6) Now, combining (3.3), (3.4), (3.5) and (3.6), it implies that g(ε), and thereforeg(ε)^n−j, are differen- tiable at 0. Moreover,

d dε

ε=0⁺

g(ε)^n−j= (n−j)g(0)^n−j−1 d dε

ε=0⁺

g(ε)

= n−j nϕ₋(1)

Sⁿ⁻¹ϕ

h_L(u) h_K(u)

h_K(u)dS_n−j−1(K, u).

This completes the proof.

Theorem 3.1 allows us to introduce the following:

Definition 3.2. For convex bodiesK, L∈ Kⁿ_o, ϕ∈Φ andj = 0,1, . . . , n−1, define the j-th Orlicz mixed quermassintegralW_ϕ,j(K, L) by

W_ϕ,j(K, L) = 1 n

Sⁿ⁻¹ϕ

h_L(u) h_K(u)

h_K(u)dS_n−j−1(K, u).

Some basic facts are observed:

(1)W_ϕ,j(K, K) =ϕ(1)W_j(K).

(2)W_ϕ,0(K, L) =V_ϕ(K, L).

(3) Ifϕ(t) =t^p,p1, thenW_ϕ,j(K, L) =W_p,j(K, L).

(4)W_ϕ,j(gK, gL) =W_ϕ,j(K, L), for all rotationsg.

(5) IfL1⊂L2, thenW_ϕ,j(K, L1)W_ϕ,j(K, L2).

(6) For allε >0,

W_j(K+_ϕε·_ϕL)/ϕ(1) =W_ϕ,j(K+_ϕε·_ϕL, K) +εW_ϕ,j(K+_ϕε·_ϕL, L).

Letϕ_i, ϕ∈Φ andi∈N. We say thatϕ_i→ϕ, provided

i→∞lim max

t∈I |ϕi(t)−ϕ(t)|= 0, for each compact intervalI⊂[0,∞).

The next lemma shows the continuity of Orlicz mixed quermassintegrals.

Lemma 3.3. Let K_i, L_i, K, L∈ Kⁿ_o andi∈N.

(1)If K_i→K, thenW_ϕ,j(K_i, L)→W_ϕ,j(K, L).

(2)If L_i→L, thenW_ϕ,j(K, L_i)→W_ϕ,j(K, L).

(3)Let ϕ_i, ϕ∈Φ. Ifϕ_i →ϕ, then W_ϕi,j(K, L)→W_ϕ,j(K, L).

Proof. Note thatK_i→K implies

S_n−j−1(K_i,·)→S_n−j−1(K,·) weakly onSⁿ⁻¹. In addition, the convergence in

ϕ(h_L/h_Ki)h_Ki →ϕ(h_L/h_K)h_K is uniform onSⁿ⁻¹. Thus, we derive (1) immediately.

Since the convergence in

ϕ(h_Li/h_K)h_K →ϕ(h_L/h_K)h_K is uniform onSⁿ⁻¹, (2) is obtained directly.

Take

r_m= min

u∈Sⁿ⁻¹h_L(u)/h_K(u) and r_M = max

u∈Sⁿ⁻¹h_L(u)/h_K(u).

Note that ϕ_i → ϕ implies ϕ_i|[rm,rM] → ϕ|[rm,rM] uniformly. Therefore, ϕ_i(h_L/h_K) → ϕ(h_L/h_K) uni- formly onSⁿ⁻¹, which implies (3).

4 The generalized Cauchy-Kubota formula

Quermassintegrals are also fundamental and useful in integral geometry. In this section, we show the probabilistic essence of Orlicz mixed quermassintegrals. The starting point is the Cauchy-Kubota formula.

Recall that for a convex body Kin Rⁿ, W_j(K) = ω_n

ω_n−j

Gn,n−j

vol_n−j(K|ξ)dσ_n,n−j(ξ), j= 1, . . . n−1.

We generalize this formula to the Orlicz setting. For this aim, the next lemma is needed, which was previously proved in [9]. Here we give a direct proof.

Lemma 4.1. Let K, L∈ K_oⁿ,ϕ∈Φ, andj= 1, . . . n−1. Then for each ξ∈G_n,j andε >0, (K+_ϕε·ϕL)|ξ=K|ξ+_ϕε·ϕL|ξ.

Proof. Letξ∈G_n,j be arbitrary but fixed, and let

S^j−1=Sⁿ⁻¹∩ξ.

For anyu∈S^j−1and Q∈ Kⁿ_o, it has

h_Q(u) =h_Q|ξ(u).

Thus, applying the definition ofK+_ϕε·ϕL tou, it gives ϕ

h_K|ξ(u) h(K+_ϕε·ϕL)|ξ(u)

+εϕ

h_L|ξ(u) h(K+_ϕε·ϕL)|ξ(u)

=ϕ(1).

On the other hand, from the definition ofK|ξ+_ϕ,εL|ξdefined inξ, it gives ϕ

h_K|ξ(u) h_K|ξ+_ϕε·ϕL|ξ(u)

+εϕ

h_L|ξ(u) h_K|ξ+_ϕε·ϕL|ξ(u)

=ϕ(1).

Hence, (K+_ϕε·ϕL)|ξandK|ξ+_ϕε·ϕL|ξ is a same convex body inξ.

Theorem 4.2 provides a probabilistic approach to define Orlicz mixed quermassintegrals.

Theorem 4.2. SupposeK, L∈ Kⁿ_o andϕ∈Φ. Then for each j= 1, . . . , n−1, W_ϕ,j(K, L) = ω_n

ω_n−j

Gn,n−j

V_ϕ^(n−j)(K|ξ, L|ξ)dσ_n,n−j(ξ),

whereV_ϕ^(n−j)(K|ξ, L|ξ)denotes the Orlicz mixed volume of the(n−j)-dimensional convex bodiesK|ξ andL|ξin the subspaceξ.

Proof. From Definition 3.2, the Cauchy-Kubota formula and Lemma 4.1, it follows that W_ϕ,j(K, L) = ϕ₋(1)

n−j ω_n ω_n−j

d dε

ε=0⁺

Gn,n−j

vol_n−j(K|ξ+_ϕε·ϕL|ξ)dσ_n,n−j(ξ).

By Theorem 3.1, the above integrand depends smoothly onε(for smallε). Hence, it gives W_ϕ,j(K, L) = ϕ₋(1)

n−j ω_n ω_n−j

Gn,n−j

d dε

ε=0⁺

vol_n−j(K|ξ+_ϕε·_ϕL|ξ)dσ_n,n−j(ξ)

= ϕ₋(1) n−j

ω_n ω_n−j

Gn,n−j

(n−j)

ϕ₋(1) V_ϕ^(n−j)(K|ξ, L|ξ)dσ_n,n−j(ξ)

= ω_n ω_n−j

Gn,n−j

V_ϕ^(n−j)(K|ξ, L|ξ)dσ_n,n−j(ξ), as desired.

Up to a constant, the quantityW_ϕ,j(K, L) is the expectation of the random variable V_ϕ^(n−j)(K| ·, L| ·) :G_n,n−j →(0,∞), ξ→V_ϕ^(n−j)(K|ξ, L|ξ),

which is defined on the probability space (G_n,n−j,B, σ_n,n−j) (where B is the Borel sigma-algebra on G_n,n−j).

Letting ϕ(t) =t^p withp1 in Theorem 4.2, it yields the formula W_p,j(K, L) = ω_n

ω_n−j

Gn,n−j

V_p^(n−j)(K|ξ, L|ξ)dσ_n,n−j(ξ).

A general representation [24, 26] of the Cauchy-Kubota formula states that for a convex bodyKinRⁿ and 1qj < n,

W_j(K) = ω_n ω_n−q

Gn,n−q

W_j−q^(n−q)(K|ξ)dσ_n,n−q(ξ),

where W_j−q^(n−q) denotes the (j−q)-th quermassintegral in the subspaceξ. Using an argument similar to that in Theorem 4.2, we can obtain the following theorem.

Theorem 4.3. SupposeK, L∈ Kⁿ_o andϕ∈Φ. Then for 1qj < n, W_ϕ,j(K, L) = ω_n

ω_n−q

Gn,n−q

W_ϕ,j−q^(n−q)(K|ξ, L|ξ)dσ_n,n−q(ξ),

whereW_ϕ,j−q^(n−q)(K|ξ, L|ξ)denotes the Orlicz mixed quermassintegral of the (n−q)-dimensional convex bodies K|ξandL|ξin the subspace ξ.

5 Inequalities for Orlicz mixed quermassintegrals

Recall that for convex bodiesK, LinRⁿandj= 0,1, . . . , n−1, Minkowski’s isoperimetric inequality (3.2) can be rewritten as

W_j(K, L) W_j(K)

W_j(L) W_j(K)

_n−j¹

, (5.1)

with equality if and only ifKandLare homothetic. The next result extends this inequality to the Orlicz setting.

Theorem 5.1. SupposeK, L∈ Kⁿ_o andϕ∈Φ. Then for each j= 0,1, . . . , n−1, W_ϕ,j(K, L)

W_j(K) ϕ

W_j(L) W_j(K)

_n−j¹ .

If ϕis strictly convex, the equality holds if and only ifK andL are dilates.

Proof. The condition onK guarantees thatW_j(K)>0. Thus, the measure h_K

nW_j(K)dS_n−j−1(K,·)

is a probability measure onSⁿ⁻¹. From the convexity ofϕcombined with Jensen’s inequality, the strict monotonicity ofϕ, together with (5.1), it follows that

W_ϕ,j(K, L)

W_j(K) = 1 nW_j(K)

Sⁿ⁻¹ϕ

h_L(u) h_K(u)

h_K(u)dS_n−j−1(K, u)

ϕ

1 nW_j(K)

Sⁿ⁻¹

h_L(u)dS_n−j−1(K, u)

=ϕ

W_j(K, L) W_j(K)

ϕ

W_j(L) W_j(K)

_n−j¹ .

Now, we verify the equality conditions. The sufficiency is easy to prove. We prove the necessity.

Suppose the equality holds. From the injectivity of ϕ, we have the equality in (5.1). So, there exist x∈Rⁿ andr >0 such thatL=rK+x. Hence,

h_L(u) =rh_K(u) +x·u,

for allu∈Sⁿ⁻¹. Sinceϕis strictly convex, by the equality condition for Jensen’s inequality, we have 1

nW_j(K)

Sⁿ⁻¹

h_L(u)

h_K(u)h_K(u)dS_n−j−1(K, u) = h_L(v) h_K(v),

forS_n−j−1(K,·)-almost allv∈Sⁿ⁻¹. Therefore, forS_n−j−1(K,·)-almost allv ∈Sⁿ⁻¹, 1

nW_j(K)

Sⁿ⁻¹

r+ x·u h_K(u)

h_K(u)dS_n−j−1(K, u) =r+ x·v h_K(v). Note that

W_j(K) = 1 n

Sⁿ⁻¹h_K(u)dS_n−j−1(K, u), and the centroid ofS_n−j−1(K,·) is at the origin. Thus, we have

0 =x· 1

nW_j(K)

Sⁿ⁻¹udS_n−j−1(K, u)

= 1

nW_j(K)

Sⁿ⁻¹x·udS_n−j−1(K, u)

= x·v h_K(v),

for S_n−j−1(K,·)-almost all v ∈ Sⁿ⁻¹. By combining the above with the fact that S_n−j−1(K,·) is not concentrated on any great subsphere of Sⁿ⁻¹, we conclude that x is just the origin, and therefore K andLare dilates.

Now, we turn to the applications of Theorem 5.1.

Lemma 5.2. Supposeϕ is a strictly convex Young function, andK, L∈ Kⁿ_o.

(1) IfK andL are dilates, then for each α, β >0,K andα·_ϕK+_ϕβ·_ϕLare dilates.

(2) Supposeα, β >0. IfK andα·ϕK+_ϕβ·ϕLare dilates, then K andLare dilates.

Proof. To prove (1), assumeL=εK for some constantε >0. LetC_S denote the class {h_K|_Sⁿ⁻¹ :K∈ K_oⁿ}.

The definition of Orlicz combinations implies that the functionh_α·ϕK+_ϕβ·ϕLis the unique solution to the equation

αϕ h_K

f

+βϕ εh_K

f

=ϕ(1), f ∈C_S.

On the other hand, it is obvious to prove that there exists a uniqueδ >0 such that αϕ

1 δ

+βϕ

ε δ

=ϕ(1), which immediately implies

αϕ h_K

h_δK

+βϕ εh_K

h_δK

=ϕ(1).

Hence,α·ϕK+_ϕβ·ϕL=δK, which concludes (1).

To prove (2), assumeα·ϕK+_ϕβ·ϕL=λK for some constantλ >0. Then for arbitraryu∈Sⁿ⁻¹, αϕ

1 λ

+βϕ

h_L(u) h_α·ϕK+_ϕβ·ϕL(u)

=ϕ(1), which implies that

ϕ

h_L(u) h_α·ϕK+_ϕβ·ϕL(u)

is constant for allu∈Sⁿ⁻¹. This and the injectivity ofϕshow thatα·ϕK+_ϕβ·ϕLandLare dilates.

Theorem 5.1 yields an Orlicz extension of the Brunn-Minkowski inequality.

Theorem 5.3. SupposeK, L∈ Kⁿ_o,ϕ∈Φ, andα, β >0. Then for each j= 0,1, . . . , n−1,

ϕ(1)αϕ

W_j(K) W_j(α·_ϕK+_ϕβ·_ϕL)

_n−j¹ +βϕ

W_j(L) W_j(α·_ϕK+_ϕβ·_ϕL)

_n−j¹ .

If ϕis strictly convex, the equality holds if and only ifK andL are dilates.

Proof. For brevity, let

K =α·ϕK+_ϕβ·ϕL.

The definition ofK necessarily implies for allu∈Sⁿ⁻¹ that ϕ(1) =αϕ

h_K(u) h_K(u)

+βϕ

h_L(u) h_K(u)

.

Hence, by the definitions ofW_j andW_ϕ,j and Theorem 5.1, we have ϕ(1)W_j(K) = αW_ϕ,j(K, K) + βW_ϕ,j(K, L)

W_j(K)

αϕ

W_j(K) W_j(K)

_n−j¹ +βϕ

W_j(L) W_j(K)

_n−j¹ . From Theorem 5.1 and Lemma 5.2, the equality conditions can be obtained immediately.

When ϕ(t) = t^p, p 1, and α, β > 0, the above inequality reduces to the L_p Brunn-Minkowski inequality

W_j(α·pK+_pβ·pL)^n−jp αW_j(K)^n−jp +βW_j(L)^n−jp . The next corollary is a weaker version of Theorem 5.3.

Corollary 5.4. SupposeK, L∈ K_oⁿ,ϕ∈Φ, andj= 0,1, . . . , n−1. Then for0< α <1, W_j(α·_ϕK+_ϕ(1−α)·_ϕL)W_j(K)^αW_j(L)^1−α.

Proof. For brevity, let

K_α=α·_ϕK+_ϕ(1−α)·_ϕL.

Since ϕ is strictly increasing and convex, by Theorem 5.3 and the arithmetic mean-geometric mean (AM-GM) inequality, we have

ϕ(1)αϕ

W_j(K) W_j(K_α)

_n−j¹

+ (1−α)ϕ

W_j(L) W_j(K_α)

_n−j¹

ϕ

α

W_j(K) W_j(K_α)

_n−j¹

+ (1−α)

W_j(L) W_j(K_α)

_n−j¹

ϕ

W_j(K)^n−jα W_j(L)^1−αn−j W_j(K_α)^n−j1

. Therefore,

W_j(K_α)W_j(K)^αW_j(L)^1−α. Ifϕ(t) =t, then the above corollary gives that for each 0< α <1,

W_j(αK+ (1−α)L)W_j(K)^αW_j(L)^1−α.

The classical difference body, DK, of a convex body K, is defined by h_DK =h_K+h_−K. The next corollary considers the Orlicz version of difference bodies.

Corollary 5.5. SupposeK∈ K_oⁿ,ϕ∈Φ, andj= 0,1, . . . , n−1. For the convex bodies Δ_ϕK=1

2·_ϕK+_ϕ1

2·_ϕ(−K) and D_ϕK=K+_ϕ(−K), there exist the inequalities

W_j(Δ_ϕK)W_j(K) and

W_j(D_ϕK)

ϕ⁻¹(ϕ(1)/2)_n−j

W_j(K).

If ϕis strictly convex, each equality holds if and only ifK is origin-symmetric.

Along with Orlicz mixed quermassintegrals, we introduce the following quantity.

Definition 5.6. For convex bodies K, L∈ Kⁿ_o andϕ∈Φ, define W_ϕ,j(K, L) = inf

λ >0 : 1 nW_j(K)

Sⁿ⁻¹ϕ

h_L(u) λh_K(u)

h_K(u)dS_n−j−1(K, u)ϕ(1)

. It can be checked that ifϕ(t) =t^p,p1, then

W_ϕ,j(K, L) = (W_p,j(K, L)/W_j(K))^1/p.

In fact,W_ϕ,j(K, L) is theOrlicz norm(see [23]) ofh_L/h_Kwith respect to the Borel probability measure h_K

nW_j(K)dS_n−j−1(K,·).