1 V(K)2 ∫ K ∫ K |u·(x−y)|dxdy, for all u∈Sn−1, (1.1) where V(K) denotes the volume of the bodyK

DONGMENG XI, LUJUN GUO, AND GANGSONG LENG

Abstract. In this paper, theLp(p≥1) mean zonoid of a convex body K is given, and we show that it is theLp centroid body of radial (n+p)th mean body ofK up to a dilation. We also establish some affine inequalities of these bodies by proving that the volume of the new bodies are decreasing under Steiner symmetrization.

1. Introduction

The notion of zonoids is basic in the Brunn-Minkowski theory of convex bodies and appear in different contexts of the mathematical literature (see e.g. [2, 12, 15, 16]). Zonoids are defined as limits of zonotopes in the Hausdorff metric, where zonotopes are Minkowski sum of segments. A zonoid Z can also be defined as a convex body whose support function is

h_Z(u) = 1 2

∫

Sⁿ⁻¹

|u·v|dµ(v), for all u∈Sⁿ⁻¹, where µis an even measure on the unit sphere Sⁿ⁻¹.

LetK ⊂Rⁿbe a convex body (compact, convex set with non-empty interior). As introduced by Zhang [18], a mean zonoid ZKe is defined by

h_ZKe (u) = 1 V(K)²

∫

K

∫

K

|u·(x−y)|dxdy, for all u∈Sⁿ⁻¹, (1.1) where V(K) denotes the volume of the bodyK.

The body ZKe is indeed a zonoid (limits of Minkowski sums of line segments). It was also shown by Zhang [18] that the volume ofV(ZKe ) satisfies

V(ZKe )≥V(ZBe _K),

where B_K is the n-ball with the same volume of K. The equality holds if and only if K is an ellipsoid.

2000Mathematics Subject Classification. 52A20, 52A40.

Key words and phrases. Lp mean zonoids,Lp centroid bodies, affine inequalities, Steiner symmetrization.

The authors would like to acknowledge the support from the National Natural Science Foundation of China (10971128), the National Natural Science Foundation of China (11271244) and Shanghai Leading Academic Discipline Project (S30104).

1

Schneider and Weil [16] introduced the notion of L_p zonoids. A finite dimensional real normed space is isometric to a subspace of L_p if and only if the polar of its unit ball is anL_p zonoid. Specially, L₂ zonoids are ellipsoids inRⁿ. For p≥1, a L_p zonoid can be defined by

h_Zp(u)^p =

∫

Sⁿ⁻¹

|u·v|^pdµ(v), for all u∈Sⁿ⁻¹, (1.2) where µ is a finite even Borel measure on the unit sphere Sⁿ⁻¹. We refer to [11, 12] for the study on this subject.

It is natural for us to consider a class of bodies ZepK named Lp mean zonoids.

Definition. Let K ⊂Rⁿ be a convex body, and let p≥1. Then, the L_p mean zonoid Ze_pK of K is defined by

h_Ze

pK(z) = ( 1

V(K)²

∫

K

∫

K

|z·(x−y)|^pdxdy )¹

p, for all z ∈Rⁿ\{0}. (1.3) The casep= 1 is just theZKe defined by Zhang [18]. We will show thatZe_pK is aL_p zonoid in Section 2.

In their paper, E. Lutwak and G. Zhang [13] introduced the L_p centroid body Γ_pK, p≥ 1, with Γ1K = ΓK. It was also shown that the volume of the polar of ΓpK is maximized if the volume of K is given. This gives an L_p version of the Blaschke-Santal´o inequality. In [11], it was proved that the volume of Γ_pK is minimized if the volume of K is given. These results have found applications in asymptotic functional analysis.

Similar to the centroid body ΓK, the L_p centroid body Γ_pK is origin dependent. In this paper, the Lp mean zonoid ZepK, which is defined by (1.3), is a translation invariant analog of the L_p centroid body. Further relationship between the L_p centroid body and theL_p mean zonoid can be seen in Section 2.

Throughout this paper, we assume that Cn,p =ωn

[2^n+pω_n+pω_2n+p ω₂²ω_p−1ω_n+p−1

]ⁿ

p,

where ω_p =π^p2/Γ(1 + ^p₂).

The main result of this paper is the following theorem. The result of the case p= 1,in our theorem, is first proved in [18].

Theorem 1. Let K ⊂ Rⁿ be a convex body, and let p≥1. Then, the volumes of Ze_pK and K satisfy the following inequality:

V(ZepK)≥Cn,pV(K), (1.4)

with equality if and only if K is an ellipsoid.

This paper is organized as follows. In Section 2, we observe that Ze_pK is a dilation of Γp(Rn+pK),where Γp(·) is theLp centroid operation, andRn+pK is the radial (n+p)th mean body of K. Section 3 contains the proofs of our main results. Note that the volume ratio V(Γ_pL)/V(L) take the minimum if and only if L is an ellipsoid (see [1]), while the volume ratio V(R_n+pK)/V(K) take the minimum if and only if K is a simplex (see [5, p. 522]), we found that Theorem 1 can not be obtained from the results above in [1] and [5]. However, it seems that we could not get a proof using the same method as in [18]. Inspired by the work of Lutwak, Yang, and Zhang [10], we prove our main theorem by utilizing the Steiner symmetrization. In Section 4, two inclusion relationships are established: one of them is between Ze_pK and Γ_pΠ^∗K, the other is between Ze_pK and Ze_p(∆K), where Π^∗K is the polar projection body of K, and ∆K = (K −K)/2. The results of the casep= 1 are first obtained in [18].

2. Preliminary

2.1. Definitions and notation. LetRⁿ denote the Euclideann-dimensional space. LetSⁿ⁻¹ denote the unit sphere, Bⁿ the unit n-ball and o the origin in Rⁿ. Denote by Kⁿ the class of convex bodies (compact, convex sets with non-empty interiors) in Rⁿ, let Kⁿo be the class of members of Kⁿ containing the origin in their interiors, and letKⁿs be the class ofo-symmetric members of Kⁿ. Ifu∈Sⁿ⁻¹, we denote by u^⊥ the (n−1)-dimensional subspace orthogonal to u, byl_u the line throughoparallel to uand byl_u(x) the line through the pointx parallel tou.

Lebesguek-dimensional measureV_kinRⁿ, k = 1, . . . , n,can be identified withk-dimensional Hausdorff measure in Rⁿ. We also generally writeV instead of V_n. Letω_n =V(Bⁿ),and thus nω_n =V_n−1(Sⁿ⁻¹).Associated with a convex bodyK is its support functionh_K defined for all x∈Rⁿ\{0} by

h_K(x) := max{x·y:y∈K}.

We shall use δ to denote the Hausdorff metric onKⁿ : If K, L∈ Kⁿ, δ(K, L) is defined by δ(K, L) = max

u∈Sⁿ⁻¹|h_K(u)−h_L(u)|, or equivalently,

δ(K, L) = min{λ:K ⊆L+λBⁿ and L⊆K+λBⁿ}. The projection body ΠK of a convex body K is defined in [14] by

h_ΠK(u) = V_n−1(K|u^⊥) = 1 2

∫

Sⁿ⁻¹

|u·v|dS_K(v),

for each u ∈Sⁿ⁻¹, where K|u^⊥ is the orthogonal projection of K on u^⊥, S_K(·) is the surface area measure ofK.

A setLis star-shaped with respect to the pointxif every line passing through xcrosses the boundary ofLat exactly two points different from x. IfLis a compact set that is star-shaped with respect to x, its radial function ρ_L(x, z) :Rⁿ\{x} →[0,∞) with respect to x is defined by

ρ_L(x, z) = max{c:x+cz∈L}, for all z ∈Rⁿ\{x}. (2.1) When x is the origin, we also denote ρ_L(o, z) by ρ_L(z) and refer to it simply as the radial function of L. By a star body we mean a compact set L whose radial function is positive and continuous. Note that this implies o∈intL.

The L_p centroid body Γ_pK of a star body K is defined by h_ΓpK(u)^p = 1

V(K)

∫

K

|u·x|^pdx= 1 (n+p)V(K)

∫

Sⁿ⁻¹

|u·v|^pρ_K(v)^n+pdv, (2.2) for all u ∈ Sⁿ⁻¹. We denote the polar body of K by K^∗. The difference body DK of K is defined by DK =K−K.

LetK ∈ Kⁿ, for all r≥0 and u∈Sⁿ⁻¹, define

E_K(r, u) ={y∈u^⊥ :|l_u(y)∩K| ≥r} and

aK(r, u) = Vn−1(EK(r, u)).

In [18]a_K(r, u) is called the restricted chord projection function ofK.It is clear thatE_K(0, u) = K|u^⊥, and a_K(0, u) =h_ΠK(u).When r > ρ_DK(u), a_K(r, u) = 0.

The following inequality can be found in [5] or [18] that V(K)≥ 1

na_K(0, u)ρ_DK(u), (2.3)

with equality if and only if K is a simplex.

Ifλ >0, Zhang (see [18, (2.2)]) proved that

∫ _∞

0

a_K(r, u)r^λdr ≤n^λ+1β(λ+ 1, n)V(K)^λ+1a_K(0, u)^−λ, (2.4) with equality if and only if K is a simplex.

LetK be a convex body and p >−1. The radial pth mean bodyR_pK is defined in [5] by ρ_RpK(z)^p = 1

V(K)

∫

K

ρ_K(x, z)^pdx, (2.5)

and it has been shown that

∫

K

ρ_K(x, u)^pdx=

∫ _∞

0

a_K(r, u)r^pdr =

∫ _ρDK(u)

0

a_K(r, u)r^pdr. (2.6) The formula

V(K) =

∫ _∞

0

a_K(r, u)dr=

∫ ρDK(u) 0

a_K(r, u)dr (2.7)

can be obtained from (2.6).

When p≥0,it has also been proved in [5] that RpK is a o-symmetric convex body.

2.2. The L_p mean zonoids. Let K be a convex body and p≥ 1, the L_p mean zonoid Ze_pK of K is defined by (1.3). We can consistently defineZe_∞K by

h_Ze

∞K(u) = max

x,y∈K|u·(x−y)|, for all u∈Sⁿ⁻¹. In fact Ze_∞K =DK. From Jensen’s inequality, it is obvious that

Ze_pK ⊆Ze_qK ⊆DK, for 1≤p≤q.

By (1.3), (2.1), the Fubini theorem, (2.5) and (2.2) h_Ze

pK(z) = ( 1

V(K)²

∫

K

∫

K

|z·(x−y)|^pdxdy )¹

p

= ( 1

V(K)²

∫

K

∫

Sⁿ⁻¹

∫ _ρK(y,v)

0

|z·v|^pr^n+p−1drdvdy )¹

p

=

( 1

(n+p)V(K)²

∫

Sⁿ⁻¹

|z·v|^p

∫

K

ρ_K(y, v)^n+pdydv )¹

p (2.8)

=

( 1

(n+p)V(K)

∫

Sⁿ⁻¹

|z·v|^pρRn+pK(v)^n+pdv )¹

p (2.9)

=

( V(Rn+pK) (n+p)V(K)

)¹

ph_Γp(Rn+pK)(z). (2.10)

From (2.10), h_Ze

pK(z) is obviously a support function, and Ze_pK =

( V(Rn+pK) (n+p)V(K)

)¹

pΓ_p(R_n+pK). (2.11)

Since R_n+pK is o-symmetric, _(n+p)V¹ _(K)ρ_Rn+pK(v)^n+p can be seen as a density function of an even Borel measure, thus ZepK is aLp zonoid from (2.9) and (1.2).

Using (2.8) and (2.6), we have the following useful formula h_Ze

pK(z) =

( 1

(n+p)V(K)²

∫

Sⁿ⁻¹

|z·v|^p

∫ _ρDK(u)

0

a_K(r, u)r^n+pdrdv )¹

p. (2.12)

3. Proof of main result

We first give some notation and definitions about Steiner symmetrization that will be used in this Section. LetK be a convex body andu∈Sⁿ⁻¹,denote byKu the image of the orthogonal projection ofK ontou^⊥.We writeℓ_u(K;y^′) :K_u →Randℓ_u(K;y^′) :K_u→Rfor the overgraph and undergraph functions ofK in the directionu; i.e.

K ={y^′+tu:−ℓ_u(K;y^′)≤t≤ℓu(K;y^′) fory^′ ∈Ku}.

Thus the Steiner symmetral SuK of K ∈ Kⁿ in direction u can be defined as the body whose orthogonal projection ontou^⊥is identical to that ofKand whose overgraph and undergraph functions are given by

ℓ_u(S_uK;y^′) =ℓ_u(S_uK;y^′) = 1

2[ℓ_u(K;y^′) +ℓ_u(K;y^′)].

Fory^′ ∈K_u,definem_y′ =m_y′(u) by m_y′(u) = 1

2[ℓ_u(K;y^′)−ℓ_u(K;y^′)].

So that the midpoint of the chordK∩l_u(y^′) isy^′+m_y′(u)u,wherel_u(y^′) is the line throughy^′ parallel tou.The length |K∩l_u(y^′)|of this chord is denote by σ_y′ =σ_y′(u).

Throughout this section, we denote x = (x^′, s) ∈ Rⁿ⁻¹ ×R, and we will usually write hK(x^′, s) rather thanhK((x^′, s)).

The following lemma will be used in the proofs of our theorems.

Lemma 3.1. ([10, Lemma 1.2]) Suppose K ∈ Kⁿo and u ∈ Sⁿ⁻¹. For y^′ ∈ relintK_u, the overgraph and undergraph functions of K in direction u are given by

ℓu(K;y^′) = min

x^′∈u^⊥{hK(x^′,1)−x^′·y^′}, (3.1a) and

ℓ_u(K;y^′) = min

x^′∈u^⊥{h_K(x^′,−1)−x^′·y^′}. (3.1b) We refer to [10] for a proof, and [1] for an application to the proof of the Lp Busemann-Petty centroid inequality.

We next show the operationZe_p :Kⁿ→ Kⁿs is continuous.

Lemma 3.2. Suppose p≥1, K_i ∈ Kⁿ, and K_i →K ∈ Kⁿ, thenZe_pK_i →Ze_pK.

Proof. Supposeu₀ ∈Sⁿ⁻¹.We will show that h_Ze

pKi(u0)→h_Ze

pK(u0).

Since Ki→K implies{Ki} are uniformly bounded, there is R >0,such thatKi ⊆RBⁿ.

By (1.3) and Minkowski’s inequality, we have

|h_Ze

pKi(u₀)−h_Ze

pK(u₀)|=( 1 V(K_i)²

∫

RBⁿ

∫

RBⁿ

1_Ki(x)1_Ki(y)|u₀·(x−y)|^pdxdy )¹

p

−( 1 V(K)²

∫

RBⁿ

∫

RBⁿ

1_K(x)1_K(y)|u₀·(x−y)|^pdxdy )¹

p

≤( 1 V(K_i)²

∫

RBⁿ

∫

RBⁿ

|1_Ki(x)1_Ki(y)−1_K(x)1_K(y)||u₀·(x−y)|^pdxdy )¹

p

+(

( 1

V(K_i)² − 1 V(K)²)

∫

RBⁿ

∫

RBⁿ

1K(x)1K(y)|u0·(x−y)|^pdxdy )¹

p. Obviously these two integrals converge to 0 whenKi →Kin the Hausdorff metric, thush_Ze

pKi(u0)→ h_Ze

pK(u₀).

Since for support functions onSⁿ⁻¹pointwise and uniform convergence are equivalent, we complete

the proof.

We show that the operatorZep :Kⁿ→ Ksⁿis GL(n) covariant in the following lemma.

Lemma 3.3. Suppose p≥1.For a convex body K ∈ Kⁿ, and a linear transformϕ∈GL(n), then Ze_p(ϕK) =ϕ(Ze_pK).

Proof. By (1.3) and the substitution x=ϕx₁, y=ϕy₁ we have h_Ze

p(ϕK)(z) = ( 1

V(ϕK)²

∫

ϕK

∫

ϕK

|z·(x−y)|^pdxdy )¹

p

= ( 1

V(ϕK)²|ϕ|²

∫

K

∫

K

|z·(ϕx−ϕy)|^pdx₁dy₁ )¹

p

= ( 1

V(K)²

∫

K

∫

K

|ϕ^tz·(x₁−y₁)|^pdx₁dy₁ )¹

p

=h_Ze

pK(ϕ^tz) =h_ϕ(Ze

pK)(z).

Thus,Ze_p(ϕK) =ϕ(Ze_pK).

The following lemma plays a key role in the proof of Theorem 1.

Lemma 3.4. Let K∈ Kⁿ, p≥1, and u∈Sⁿ⁻¹.If z₁^′, z₂^′ ∈u^⊥,then h_Ze

p(SuK)(z^′₁+z₂^′

2 ,1)≤ 1 2h_Ze

pK(z₁^′,1) +1 2h_Ze

pK(z₂^′,−1), (3.2a) and

h_Ze

p(SuK)(z₁^′ +z₂^′

2 ,−1)≤ 1 2h_Ze

pK(z₁^′,1) + 1 2h_Ze

pK(z₂^′,−1). (3.2b) Equality in (3.2a) or (3.2b) implies that all of the chords of K parallel to u, have midpoints that lie in a hyperplane.

Proof. From the definition ofL_p mean zonoid, we have h_Ze

pK(z₁^′,1) = ( 1

V(K)²

∫

K

∫

K

|(z₁^′,1)·(x−y)|^pdxdy )¹

p

= ( 1

V(K)²

∫

Ku

∫ _my′+¹

2σ_y′

m_y′−¹₂σ_y′

∫

Ku

∫ _mx′+¹

2σ_x′

m_x′−¹₂σ_x′

|(z₁^′,1)·((x^′, s1)−(y^′, s2))|^pds1dx^′ds2dy^′ )¹

p

= ( 1

V(K)²

∫

Ku

∫ _my′+¹

2σ_y′

my′−¹₂σy′

∫

Ku

∫ _mx′+¹

2σ_x′

mx′−¹₂σx′

|z^′₁·(x^′−y^′) +s1−s2|^pds1dx^′ds2dy^′ )¹

p

= ( 1

V(K)²

∫

Ku

∫ ¹

2σ_y′

−¹₂σy′

∫

Ku

∫ ¹

2σ_x′

−¹₂σx′

|z₁^′ ·(x^′−y^′) +t1−t2+mx^′ −my^′|^pdt1dx^′dt2dy^′ )¹

p

=

( 1

V(S_uK)²

∫

SuK

∫

SuK

|z₁^′ ·(x^′−y^′) +t₁−t₂+m_x′ −m_y′|^pdt₁dx^′dt₂dy^′ )¹

p,

by making the change of variables t1=−mx^′+s1, t2 =−my^′+s2,and h_Ze

pK(z₂^′,−1) = ( 1

V(K)²

∫

K

∫

K

|(z₂^′,−1)·(x−y)|^pdxdy )¹

p

= ( 1

V(K)²

∫

Ku

∫ _my′+¹

2σ_y′

m_y′−¹₂σ_y′

∫

Ku

∫ _mx′+¹

2σ_x′

m_x′−¹₂σ_x′

|(z₂^′,−1)·((x^′, s1)−(y^′, s2))|^pds1dx^′ds2dy^′ )¹

p

= ( 1

V(K)²

∫

Ku

∫ _my′+¹

2σ_y′

my′−¹₂σy′

∫

Ku

∫ _mx′+¹

2σ_x′

mx′−¹₂σx′

|z₂^′ ·(x^′−y^′)−s1+s2|^pds1dx^′ds2dy^′ )¹

p

= ( 1

V(K)²

∫

Ku

∫ ¹

2σ_y′

−¹₂σy′

∫

Ku

∫ ¹

2σ_x′

−¹₂σx′

|z₂^′ ·(x^′−y^′) +t1−t2−mx^′+my^′|^pdt1dx^′dt2dy^′ )¹

p

=

( 1

V(S_uK)²

∫

SuK

∫

SuK

|z₂^′ ·(x^′−y^′) +t₁−t₂−m_x′+m_y′|^pdt₁dx^′dt₂dy^′ )¹

p,

by making the change of variables t1=mx^′−s1, t2 =my^′ −s2. Then, by Minkowski’s inequality, we have

2h_Ze

p(SuK)(z^′₁+z₂^′

2 ,1) = 2

( 1

V(SuK)²

∫

SuK

∫

SuK

|(z₁^′ +z^′₂

2 ,1)·(x−y)|^pdxdy )¹

p

=

( 1

V(SuK)²

∫

SuK

∫

SuK

|(z₁^′ +z^′₂)·(x^′−y^′) + 2t₁−2t₂|^pdt₁dx^′dt₂dy^′ )¹

p

≤( 1 V(SuK)²

∫

SuK

∫

SuK

|z^′₁·(x^′−y^′) +t₁−t₂+m_x′ −m_y′|^pdt₁dx^′dt₂dy^′ )¹

p

+

( 1

V(S_uK)²

∫

SuK

∫

SuK

|z₂^′ ·(x^′−y^′) +t₁−t₂−m_x′+m_y′|^pdt₁dx^′dt₂dy^′ )¹

p

=h_Ze

pK(z^′₁,1) +h_Ze

pK(z₂^′,−1).

Thus we established (3.2a), and (3.2b) can be obtained by the same way.

Since the inequality is obtained by Minkowski’s inequality, it takes equality in (3.2a) or (3.2b) if and only if there isλ≥0,such that

z₁^′ ·(x^′−y^′) +t1−t2+m_x′−m_y′ =λ(z^′₂·(x^′−y^′) +t1−t2−m_x′ +m_y′), for all (x^′, t₁),(y^′, t₂)∈K. This is equivalent to

(z₁^′ −λz₂^′)·(x^′−y^′) + (1 +λ)(m_x′−m_y′) = (λ−1)(t1−t2), (3.3) for all (x^′, t₁),(y^′, t₂)∈K.

If we fix x^′, y^′ and change t1, t2 in (3.3) such that (x^′, t1),(y^′, t2) ∈ K, then the left of (3.3) will not change, this implies that λ= 1.Thus, equality in (3.2a) or (3.2b) implies all of the chords ofK

parallel tou, have midpoints that lie in a hyperplane.

The theorems will be proved using the following lemma.

Lemma 3.5. Let K∈ Kⁿ, p≥1, and u∈Sⁿ⁻¹,then

Ze_p(S_uK)⊆S_u(Ze_pK). (3.4)

If the inclusion is an identity then all of the chords of K parallel to u, have midpoints that lie in a hyperplane.

Proof. Supposey^′ ∈relint(Ze_pK)_u. By Lemma 3.1 there exist z₁^′ =z₁^′(y^′) and z₂^′ =z^′₂(y^′) inu^⊥ such that

ℓ_u(Ze_pK;y^′) =h_Ze

pK(z₁^′,1)−z^′₁·y^′, ℓ_u(Ze_pK;y^′) =h_Ze

pK(z^′₂,−1)−z₂^′ ·y^′. By (3.2) and (3.1), we have

ℓu(Su(ZepK);y^′) = 1

2ℓu(ZepK;y^′) +1

2ℓ_u(ZepK;y^′)

= 1 2(h_Ze

pK(z^′₁,1)−z₁^′ ·y^′) +1 2(h_Ze

pK(z₂^′,−1)−z₂^′ ·y^′)

= 1 2h_Ze

pK(z₁^′,1) +1 2h_Ze

pK(z^′₂,−1)−(1 2z₁^′ +1

2z^′₂)·y^′

≥h_Ze

p(SuK)(z₁^′ +z₂^′

2 ,1)−(1 2z₁^′ +1

2z₂^′)·y^′

≥ min

x^′∈u^⊥{h_Ze

p(SuK)(x^′,1)−x^′·y^′}

=ℓu(Zep(SuK);y^′), and

ℓ_u(Su(ZepK);y^′) = 1

2ℓu(ZepK;y^′) +1

2ℓ_u(ZepK;y^′)

= 1 2 (

h_Ze

pK(z₁^′,1)−z^′₁·y^′ )

+1 2 (

h_Ze

pK(z^′₂,−1)−z^′₂·y^′ )

= 1 2h_Ze

pK(z₁^′,1) +1 2h_Ze

pK(z₂^′,−1)−(1 2z₁^′ +1

2z^′₂)·y^′

≥h_Ze

p(SuK)(z₁^′ +z₂^′

2 ,−1)−(1 2z₁^′ +1

2z^′₂)·y^′

≥ min

x^′∈u^⊥{h_Ze

p(SuK)(x^′,−1)−x^′·y^′}

=ℓ_u(Ze_p(S_uK);y^′).

If the inclusion is an identity, it must take equality in both (3.2a) and (3.2b), this implies all of the chords ofK parallel tou,have midpoints that lie in a hyperplane.

Proof of Theorem 1. Choose a sequence of directions {ui} such that the sequence{Ki}defined by

Ki+1=SuiKi, K0=K converges toBK,whereBK is the n-ball such that V(K) =V(BK).

Since the Steiner transform keeps the volume, by Lemma 3.5 and Lemma 3.2 we have V(Ze_pK)≥V(Ze_pB_K).

IfK is an ellipsoid, then V(Ze_pK) =V(Ze_pB_K) according to Lemma 3.3.

Conversely, if V(Ze_pK) =V(Ze_pB_K),the inclusion in (3.4) must be identity for allu∈Sⁿ⁻¹.This shows that all of the chords ofKparallel tou, have midpoints that lie in a hyperplane for allu∈Sⁿ⁻¹,

and thus K is an ellipsoid.

Thus, we have

V(ZepK)≥V(ZepBK), (3.5)

whereBK is the n-ball with the same volume of K, with equality if and only ifK is an ellipsoid.

We claim that

[V(ZepBK) ω_n

]¹

n = ( 1

V(B_K)²

∫

BK

∫

BK

|u·(x−y)|^pdxdy )¹

p

=

( (n+p)ω_n+p nω2ωp−1ωnV(K)²

∫

BK

∫

BK

|x−y|^pdxdy )¹

p.

Since h_Ze

pBK(u) is a constant independent of u, it is clear that ZepBK is a n-ball, we get the first equality. The second equality is obtained by integral

∫

Sⁿ⁻¹

|u·(x−y)|^pdu= (n+p)ω_n+p

ω2ωp−1 |x−y|^p.

By using the spherical polar coordinates, (2.1), the Fubini theorem, and (2.6),

∫

BK

∫

BK

|(x−y)|^pdxdy =

∫

BK

∫

Sⁿ⁻¹

∫ _ρBK(y,u)

0

r^n+p−1drdudy

= 1

n+p

∫

Sⁿ⁻¹

∫

BK

ρBK(y, u)^n+pdydu

= 1 n+p

∫

Sⁿ⁻¹

∫ _ρDBK(u)

0

aBK(r, u)r^n+pdrdu

= ωn−1V(K)^2n+pn n+p

∫

Sⁿ⁻¹

∫ ₂

0

(1−(r

2)²)ⁿ⁻²¹r^n+pdrdu

= 2^n+pnωnωn−1

n+p β(n+ 1

2 ,n+p+ 1

2 )V(K)^2+pn

= 2^n+pnω_nω_2n+p (n+p)ω2ωn+p−1

V(K)^2+np. Combining these together, we have

[V(Ze_pB_K) ωn

]¹

n =

[2^n+pω_n+pω_2n+p ω²₂ωp−1ωn+p−1

]¹

pV(K)¹ⁿ. (3.6)

From (3.5) and (3.6), we get (1.4).

4. Further results In this section, we assume that

C_K(n, p) =n^n+p+1p β(n+p+ 1, n)^1pV(K)^n+p−1p V(Π^∗K)^1p. Theorem 4.1. Let K∈ Kⁿ and p≥1,then

Ze_pK⊆C_K(n, p)Γ_pΠ^∗K, with equality if and only ifK is a simplex.

Proof. Letu∈Sⁿ⁻¹.We will prove h_Ze

pK(u)≤C_K(n, p)h_ΓpΠ∗K(u), (4.1) with equality for some uif and only if K is a simplex.

By (2.12) and (2.4), we have h_Ze

pK(u)≤(

n^n+pβ(n+p, n+ 1)V(K)^n+p−1

∫

Sⁿ⁻¹

|uv|^pa_K(0, v)^−n−pdv )¹

p. (4.2)

By (2.2) and the fact thatρΠ^∗K(v) =aK(0, v)⁻¹, hΓpΠ^∗K(u) =

( 1

(n+p)V(Π^∗K)

∫

Sⁿ⁻¹

|uv|^paK(0, v)^−n−pdv )¹

p. (4.3)

Then, (4.2) and (4.3) imply (4.1), with equality if and only if K is a simplex.

The following lemma is crucial in the proof of Theorem4.3.