# The Characteristic Properties of the MinimalLp-Mean Width

Authors: Tongyi Ma
Journal: Journal of Function Spaces
 (2017)

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## Abstract

Giannopoulos proved that a smooth convex bodyK has minimal mean width position if and only if the measure hK(u)σ(du), supported on Sn-1, is isotropic. Further, Yuan and Leng extended the minimal mean width to the minimal Lp-mean width and characterized the minimal position of convex bodies in terms of isotropicity of a suitable measure. In this paper, we study the minimal Lp-mean width of convex bodies and prove the existence and uniqueness of the minimal Lp-mean width in its SL(n) images. In addition, we establish a characterization of the minimal Lp-mean width, conclude the average Mp(K) with a variation of the minimal Lp-mean width position, and give the condition for the minimum position of Mp(K).


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## Body

## 1. Introduction

LetLn(Rn) denote the space of linear operators from Rn to Rn and SL(n)={T∈Ln(Rn):det⁡(T)=1}. Suppose K is a convex body in Rn; then {TK:T∈SL(n)} is the family of its positions. In [1] it was shown that for many natural functionals of the form T↦f(TK), T∈SL(n), the solution T0 of the problem min⁡{f(TK):T∈SL(n)} is isotropic with respect to an appropriate measure depending on f. The purpose of this note is to provide applications of this point of view in the case of the mean width functional T↦w(TK) under various constraints.Recall that the width ofK in the direction of u∈Sn-1 is defined by w(K,u)=hK(u)+hK(-u), where hK(x) is the support function of convex K. The mean width of K is given by (1)wK=∫Sn-1wK,uσdu=2∫Sn-1hKuσdu,where σ is the rotationally invariant probability measure on the unit sphere Sn-1.We say thatK has minimal mean width if w(K)≤w(TK) for every T∈SL(n). In [1], the authors show that a smooth enough convex body (that is, hK is twice continuously differentiable) is in minimal mean width position if and only if the measure hK(u)σ(du), supported on Sn-1, is isotropic.Theorem 1.
A smooth enough convex bodyK in Rn has minimal mean width if and only if (2)∫Sn-1u,θ2hKuσdu=wK2n,foreveryθ∈Sn-1.Moreover, if U∈SL(n) and UK has minimal mean width, we must have U∈O(n), where O(n) denotes rotation transformation group.Yuan et al. in [2] introduced the notion of the minimal Lp-mean width of convex body. Let K be a convex body in Rn and p≥1; the Lp-width of K in the direction of u∈Sn-1 is defined by wp(K,u)=hKp(u)+hKp(-u). The Lp-width function wp(K,·) is translation invariant; therefore we may assume that o∈intK. The Lp-mean width of K is given by (3)wpK=∫Sn-1wpK,uσdu=2∫Sn-1hKpuσdu.Ifwp(K)≤wp(TK) for every T∈SL(n), then we say that K has minimal Lp-mean width. The following isotropic characterization of the minimal Lp-mean width position was proved in [2].Theorem 2.
A smooth enough convex bodyK in Rn has minimal Lp-mean width if and only if (4)∫Sn-1u,θ2hKpuσdu=wpK2n,foreveryθ∈Sn-1.Moreover, if U∈SL(n) and UK has minimal p-mean width, we must have U∈O(n).It is easily seen that theLp-mean width belongs to the Lp-Brunn-Minkowski theory. The Lp-Brunn-Minkowski theory is far more general than the classical Brunn-Minkowski theory; we refer the reader to [3–25].The aim of this article is to study the characterization of the minimalLp-mean width, which enriches the theory of Lp-mean width of convex bodies. In Section 3, we discuss the continuity of the Lp-mean width of convex bodies. In Section 4, we mainly demonstrate that the body K has a unique SL(n) image with minimal Lp-mean width. In view of this fact, we define the minimal Lp-mean width of K by (5)w-pK=min⁡wpTK:T∈SLn.In Section 5, we prove the following equivalent conditions with the minimal Lp-mean width of convex bodies.Theorem 3.
SupposeK is a smooth enough convex body in Rn, p≥1, and T0∈SL(n). Then the following assertions are equivalent:(1)
w-p(K)=wp(T0K).(2)
The measurewp(T0K,·) is isotropic on Sn-1.(3)
For allx∈Rn, the transformation T0 satisfies (6)x2∫Sn-1T0-tu-n+phKpuσdu=n∫Sn-1x,T0-tu2T0-tu-n+p+2hKpuσdu.In Section6, we consider the average (7)MpK=∫Sn-1xKpσdxof the norm ·K on Sn-1. Thus we define Mp∗(K)=Mp(K∘) and obtain the following condition for the minimum position.Theorem 4.
LetK be a symmetric convex body in Rn and assume that Mp(K)Mp∗(K)≤Mp(TK)Mp∗(TK) for every T∈SL(n). Then (8)1MpK∫Sn-1u,θ2uKpσdu=1Mp∗K∫Sn-1u,θ2uK∘pσdu,foreveryθ∈Sn-1.In addition, we also get the following condition for the minimum position.Theorem 5.
LetK be a symmetric convex body in Rn satisfying a(K)=1 and Mp(K)≤Mp(TK) for every T∈GL(n) with a(TK)=1. Then, for every θ∈Sn-1 we can find contact points x1, x2 of K and B such that (9)1+px1,θ2≤n+pMpK∫Sn-1uKpu,θ2σdu≤1+px2,θ2,where the condition a(TK)=1 means that TK⊆B but there exist contact points of TK and B (see [1]).Please see the next section for above interrelated notations, definitions, and their background materials.


## 2. Preliminaries

### 2.1. Notations

The setting will be the Euclideann-space Rn. As usual, 〈x,y〉 denotes the standard inner product of x and y in Rn; B={x∈Rn:〈x,x〉≤1} and Sn-1=∂B denotes the unit ball and unit sphere in Rn, respectively. The volume of B is ωn=πn/2/Γ(1+n/2). We often use |·| to denote the standard Euclidean norm, on occasion the total mass of a measure, and the absolute value of the determinant of an n×n matrix. For brevity, we write 〈x〉=x/|x|, for x∈Rn∖{o}.Given a compact convex setK in Rn, its support function hK(·)=h(K,·):Rn→R is defined by (10)hKx=max⁡x,y:y∈K.The definition immediately gives that, for T∈GL(n), (11)hTKx=hKTtx,x∈Rn.As usual, write Kon for the class of convex bodies in Rn that contain the origin in their interiors. Kon is often equipped with the Hausdorff metric δH, which is defined by (12)δHK1,K2=max⁡hK1u-hK2u:u∈Sn-1=hK1-hK2∞,forK1,K2∈Kon.ForK∈Kon, define RK,rK∈(0,∞) by (13)RK=maxu∈Sn-1hKu,rK=minu∈Sn-1hKu.A setK⊂Rn is said to be a star body about the origin, if the line segment from the origin to any point x∈K is contained in K and K has continuous and positive radial function ρK(·). Here, the radial function of K, ρK:Rn∖{o}→[0,∞), is defined by (14)ρKx=ρK,x=max⁡λ≥0:λx∈K,x∈Rn∖o.We write Son for the class of star bodies about the origin o in Rn. Son is often equipped with the dual Hausdorff metric δ~H, which is defined by (15)δ~HK,L=max⁡ρKu-ρLu:u∈Sn-1≔ρKu-ρLu∞,forK,L∈Son.If convex bodyK contains the origin o as its interior point, then the Minkowski functional ·K of K is defined by (16)xK=minλ>0:x∈λK.In this case, xK=hK∘(x), where K∘ denotes the polar set of K, which is defined by (17)K∘=x∈Rn:x,y≤1∀y∈K.IfK∈Kon and any T∈GL(n), then (18)TK∘=T-tK∘.In addition, we easily see that if K∈Kon, then the support and radial functions of K∘, the polar body of K, are defined, respectively, by (19)hK∘=ρK-1,ρK∘=hK-1.


### 2.2. Linear Operators

ForT∈Ln(Rn), Tt and T denote the transpose and norm of T, respectively. For K∈Kon, let d(K) denote its maximal principal radius.Two facts are in order (see [26]). First, T∈Ln(Rn) is nondegenerated, if and only if the ellipsoid TB is nondegenerated. Second, for T∈Ln(Rn), since (20)T=supu∈Sn-1Tu=supu∈Sn-1Ttu=Tt,it follows that (21)dTB=supu∈Sn-1hTBu=supu∈Sn-1Ttu=supu∈Sn-1Tu=supu∈Sn-1hTtBu=dTtB.Letdn(T1,T2)=T1-T2, for T1,T2∈Ln(Rn). Then the metric space (Ln(Rn),dn) is complete. Since Ln(Rn) is of finite dimension, a set in (Ln(Rn),dn) is compact, if and only if it is bounded and closed.Lemma 6 (see [26]).
Suppose{Ti}i∈N⊂SL(n). Then Ti→∞⇔Ti-1→∞.Thus,{Ti}i∈N is bounded from above, if and only if {Ti-1}i∈N is bounded from above.Lemma 7 (see [26]).
Suppose{Ti}i∈N⊂SL(n) and Ti→T0⊂SL(n) with respect to dn. Then(1)
TitB→T0tB with respect to δH.(2)
Ti-1→T0-1 with respect to dn.(3)
TiB→T0B with respect to δ~H.


### 2.3. Notion of Isotropy of Measures

For further discussion, we introduce the important notion of isotropy of measures. A nonnegative Borel measureμ on Sn-1 is said to be isotropic if (22)∫Sn-1u,v2dμu=μn,∀v∈Sn-1.Here, |μ| denotes the total mass of μ. The definition immediately yields (23)∫Sn-1ui2dμu=μn,where ui denotes the ith component of the coordinate of u. For nonzero x∈Rn, the notation x⊗x represents the linear operator of the rank 1 on Rn that takes y to 〈x,y〉x. It immediately gives the fact that tr(x⊗x)=|x|2. Equivalently, μ is isotropic if (24)∫Sn-1u⊗udμu=μnIn,where In denotes the identity operator on Rn. For more information about the isotropy, we refer to [1, 27–29].The following fact will be needed.Lemma 8 (see [30]).
Suppose thatμ is a probability measure on a space X and f:X→I⊂R is a μ-integrable function, where I is a possibly infinite interval. Jensen’s inequality states that if φ:I→R is a convex function, then (25)∫Xφfxdμx≥φ∫Xfxdμx.If φ is strictly convex, equality holds if and only if f(x) is constant for μ, with almost all x∈X.


## 2.1. Notations

The setting will be the Euclideann-space Rn. As usual, 〈x,y〉 denotes the standard inner product of x and y in Rn; B={x∈Rn:〈x,x〉≤1} and Sn-1=∂B denotes the unit ball and unit sphere in Rn, respectively. The volume of B is ωn=πn/2/Γ(1+n/2). We often use |·| to denote the standard Euclidean norm, on occasion the total mass of a measure, and the absolute value of the determinant of an n×n matrix. For brevity, we write 〈x〉=x/|x|, for x∈Rn∖{o}.Given a compact convex setK in Rn, its support function hK(·)=h(K,·):Rn→R is defined by (10)hKx=max⁡x,y:y∈K.The definition immediately gives that, for T∈GL(n), (11)hTKx=hKTtx,x∈Rn.As usual, write Kon for the class of convex bodies in Rn that contain the origin in their interiors. Kon is often equipped with the Hausdorff metric δH, which is defined by (12)δHK1,K2=max⁡hK1u-hK2u:u∈Sn-1=hK1-hK2∞,forK1,K2∈Kon.ForK∈Kon, define RK,rK∈(0,∞) by (13)RK=maxu∈Sn-1hKu,rK=minu∈Sn-1hKu.A setK⊂Rn is said to be a star body about the origin, if the line segment from the origin to any point x∈K is contained in K and K has continuous and positive radial function ρK(·). Here, the radial function of K, ρK:Rn∖{o}→[0,∞), is defined by (14)ρKx=ρK,x=max⁡λ≥0:λx∈K,x∈Rn∖o.We write Son for the class of star bodies about the origin o in Rn. Son is often equipped with the dual Hausdorff metric δ~H, which is defined by (15)δ~HK,L=max⁡ρKu-ρLu:u∈Sn-1≔ρKu-ρLu∞,forK,L∈Son.If convex bodyK contains the origin o as its interior point, then the Minkowski functional ·K of K is defined by (16)xK=minλ>0:x∈λK.In this case, xK=hK∘(x), where K∘ denotes the polar set of K, which is defined by (17)K∘=x∈Rn:x,y≤1∀y∈K.IfK∈Kon and any T∈GL(n), then (18)TK∘=T-tK∘.In addition, we easily see that if K∈Kon, then the support and radial functions of K∘, the polar body of K, are defined, respectively, by (19)hK∘=ρK-1,ρK∘=hK-1.


## 2.2. Linear Operators

ForT∈Ln(Rn), Tt and T denote the transpose and norm of T, respectively. For K∈Kon, let d(K) denote its maximal principal radius.Two facts are in order (see [26]). First, T∈Ln(Rn) is nondegenerated, if and only if the ellipsoid TB is nondegenerated. Second, for T∈Ln(Rn), since (20)T=supu∈Sn-1Tu=supu∈Sn-1Ttu=Tt,it follows that (21)dTB=supu∈Sn-1hTBu=supu∈Sn-1Ttu=supu∈Sn-1Tu=supu∈Sn-1hTtBu=dTtB.Letdn(T1,T2)=T1-T2, for T1,T2∈Ln(Rn). Then the metric space (Ln(Rn),dn) is complete. Since Ln(Rn) is of finite dimension, a set in (Ln(Rn),dn) is compact, if and only if it is bounded and closed.Lemma 6 (see [26]).
Suppose{Ti}i∈N⊂SL(n). Then Ti→∞⇔Ti-1→∞.Thus,{Ti}i∈N is bounded from above, if and only if {Ti-1}i∈N is bounded from above.Lemma 7 (see [26]).
Suppose{Ti}i∈N⊂SL(n) and Ti→T0⊂SL(n) with respect to dn. Then(1)
TitB→T0tB with respect to δH.(2)
Ti-1→T0-1 with respect to dn.(3)
TiB→T0B with respect to δ~H.


## 2.3. Notion of Isotropy of Measures

For further discussion, we introduce the important notion of isotropy of measures. A nonnegative Borel measureμ on Sn-1 is said to be isotropic if (22)∫Sn-1u,v2dμu=μn,∀v∈Sn-1.Here, |μ| denotes the total mass of μ. The definition immediately yields (23)∫Sn-1ui2dμu=μn,where ui denotes the ith component of the coordinate of u. For nonzero x∈Rn, the notation x⊗x represents the linear operator of the rank 1 on Rn that takes y to 〈x,y〉x. It immediately gives the fact that tr(x⊗x)=|x|2. Equivalently, μ is isotropic if (24)∫Sn-1u⊗udμu=μnIn,where In denotes the identity operator on Rn. For more information about the isotropy, we refer to [1, 27–29].The following fact will be needed.Lemma 8 (see [30]).
Suppose thatμ is a probability measure on a space X and f:X→I⊂R is a μ-integrable function, where I is a possibly infinite interval. Jensen’s inequality states that if φ:I→R is a convex function, then (25)∫Xφfxdμx≥φ∫Xfxdμx.If φ is strictly convex, equality holds if and only if f(x) is constant for μ, with almost all x∈X.


## 3. The Continuity of theLp-Mean Width

UsingKi→K equivalent to hKi→hK uniformly on Sn-1, together with the convexity of ϕ(t)=tp and definition (3), we immediately obtain the following.Lemma 9.
SupposeK,Ki∈Kon and p,pj≥1, i,j∈N. If Ki→K and pj→p, then (26)limi,j→∞wpjKi=wpK.Fixing a convex bodyK, the continuity of the Lp-mean width wp with respect to T∈SL(n) is contained in the following.Lemma 10.
SupposeK∈Kon, p≥1, and T0,Ti∈SL(n). If Ti→T0, then (27)limTi→T0wpTiK=wpT0K.Proof.
From (2) of Lemma7, we have the following implications:(28)Ti-T0⟶0⇓Ti-1-T0-1⟶0⇓Ti-1u⟶T0-1u,pointwise  convergence  for  everyu∈Sn-1⇕Ti-1B⟶T0-1B,with  respect  toδH⇕hTi-1Bu⟶hT0-1Bu,uniformly  for  everyu∈Sn-1.From the above, (35), and Lemma 9, we get limTi→T0wp(TiK)=wp(T0K).Lemma 11.
SupposeK∈Kon and p≥1. Then (29)minT∈SLnT→∞⁡wpTK=∞.Proof.
Without loss of generality, we writeT in the form T=O1AO2t, where A is an n×n diagonal matrix, with det⁡(A)=1 and positive diagonal elements a1,…,an, and O1,O2 are n×n orthogonal matrices. For any y∈Rn, let y1 denote the l1-norm of y, u=(u1,…,un)∈Sn-1 with ϵu=min1≤i≤n|ui|>ϵ>0. Recall that there exists a positive C such that |y|≥Cy1, and ∑i=1n‍ai≥max1≤i≤nai=A=T. Then from the definition of Lp-mean width, Jensen’s inequality (Lemma 8, see [30, 31]), and (13), it follows that (30)wpTK=2∫Sn-1hKpAO2uσdu≥2∫Sn-1hKAuσdup=2∫Sn-1Aun+1hKvσdvpv=AuAu≥2∫Sn-1Cn+1Au1n+1hKvσdvp=2∫Sn-1Cn+1∑i=1naiuin+1hKvσdvp>2∫Sn-1ϵCn+1∑i=1nain+1hKvσdvpϵ∈0,ϵu⊆0,1≥2ϵCn+1rKTn+1p.It immediately yields (31)limT∈SLnT→∞wpTK=∞.


## 4. The MinimalLp-Mean Width

In order to demonstrate the existence and uniqueness of minimalLp-mean width, we begin by proving some lemmas.Lemma 12.
SupposeK∈Kon, p≥1, and T∈GL(n). Then (32)wpTK=2T-t∫Sn-1hTBn+puhKpvσdv,v=T-1u∈Sn-1.Proof.
From (3) and (11), we have (33)wpTK=∫Sn-1hKpTtuσdu=2T-t∫Sn-1Ttun+phKpTtuσdTtu=2T-t∫Sn-1Ttun+phKpvσdv=2T-t∫Sn-1hTBn+puhKpvσdv.We conclude the proof.ForT∈GL(n), from (18) and (19), we have (34)hTBu=Ttu=T-tTtu-1=T-tv-1=hT-1B-1v=ρTtBv.Therefore, we can get (35)wpTK=2T-t∫Sn-1hT-1B-n+pvhKpvσdv=2T-t∫Sn-1ρTtBn+pvhKpvσdv.Given an origin-symmetric ellipsoidE in Rn, let dE=max⁡{hE(u):u∈Sn-1}. Then there exists a vE∈Sn-1 such that, for all u∈Sn-1, (36)dEu,vE≤hEu.Lemma 13.
SupposeK∈Kon, p≥1, and T∈SL(n). Then (37)dTB≤wpTK/2rKp1/n+pminv∈Sn-1∫Sn-1u,vσdv.Proof.
From Lemma12, (36), and Jensen’s inequality, together with (13), we have (38)wpTK=2∫Sn-1hTBn+puhKpvσdv≥2∫Sn-1dTKu,vTKn+phKpvσdv≥2rKp∫Sn-1dTKu,vTKn+pσdv≥2rKp∫Sn-1dTKu,vTKσdvn+p≥2rKpdTK∫Sn-1u,vTKσdvn+p≥2rKpdTKn+pminv∈Sn-1∫Sn-1u,vσdvn+p.
We easily see that(39)minv∈Sn-1∫Sn-1u,vσdv>0.
Indeed, it is known that there exists a unit vectorv0∈Sn-1 such that ∫Sn-1‍|〈u,v0〉|σ(dv)=minv∈Sn-1∫Sn-1‍|〈u,v〉|σ(dv). Choose u∈Sn-1 which is not orthogonal to v0. Then the statement follows by continuity. Accordingly, the desired inequality is derived.With the previous lemmas in hand, we can show that the minimalLp-mean width is well-defined. Our purpose is to find necessary and sufficient conditions for a body K to have minimal Lp-mean width. We assume for simplicity that hK is twice continuously differentiable (we then say that K is smooth enough).Theorem 14.
SupposeK∈Kon and any T∈SL(n). Then, modulo orthogonal transformations, there exists a unique solution to the minimization problem (40)minT∈SLnwpTK.Proof.
Note that the proof is based on the idea of Zou and Xiong [26, 32]. Let {Tk}k∈N⊂SL(n) be a minimizing sequence for the problem; that is, (41)limk→∞wpTkK=inf⁡wpTK:T∈SLn.Note that (42)inf⁡wpTK:T∈SLn≤wpK<∞,which implies {wp(TkK)}k∈N; therefore {(wp(TkK)/2rKp)1/(n+p)}k∈N is bounded from above. So, by Lemma 13, {TkB}k∈N is bounded with respect to the Hausdorff metric. From the Blaschke selection theorem, {TkB}k∈N has a convergent subsequence {TkjB}j∈N that converges to a body E. Since volume functional is continuous with respect to the Hausdorff metric, and V(TkjK)=ωn for each j, it yields that V(E)=ωn; since the convergence of {TkjB}j∈N is equivalent to the uniform convergence of {hTkjB}j∈N on Sn-1, and hTkjB(u)=hTkjB(-u) for all u∈Sn-1, it yields that hE(u)=hE(-u) for all u∈Sn-1. Thus, E is a nondegenerated origin-symmetric ellipsoid.
Consequently, there exists a transformationT0∈SL(n) such that E=T0B. This demonstrates the existence of solutions to the considered problem.
Now, we prove the uniqueness by contradiction. Assume thatT1,T2∈SL(n) both solve the considered minimization problem. Let E1=T1B and E2=T2B. It is known that each T∈SL(n) can be represented in the form T=PQ, where P is symmetric, positive definite and Q is orthogonal. So, without loss of generality, we may assume that T1 and T2 are symmetric and positive definite.
The Minkowski inequality for symmetric positive definite matrices shows that(43)det⁡T1+T221/n>12det⁡T11/n+12det⁡T21/n=1.Let (44)T3=det⁡T1+T22-1/nT1+T22.Then T3∈SL(n), and, for all u∈Sn-1, (45)hE3u=hT3Bu<h1/2T1+T2Bu=T1u+T2u2≤T1u+T2u2=12hT1Bu+12hT2Bu.
Using (32) and (45) and noticing that if a,b∈R and p>0, then (|a|+|b|)p≤Cp(|a|p+|b|p) (where Cp=1 if 0<p≤1, and Cp=2p-1 if p>1), we have (46)wpT3K=2∫Sn-1hT3Bn+pvhKnvσdv<2∫Sn-112hT1Bu+12hT2Bun+phKpvσdv=∫Sn-1hT1Bn+puhKpvσdv+∫Sn-1hT2Bn+puhKpvσdv=12wϕT1K+12wϕT2K=wϕT1K=wϕT2K.Namely, (47)wϕT3K<wϕT1K=wϕT2K.
However, by the previous assumption onT1 and T2, we have (48)wϕT3K≥wϕT1K=wϕT2K,which is a contradiction. The proof is complete.In view of Theorem14, naturally, we introduce the following.Definition 15.
SupposeK∈Kon and p≥1. The quantity (49)w-pK=min⁡wpTK:T∈SLnis called the minimal Lp-mean width of the convex body K with respect to p.


## 5. The Characterization of the MinimalLp-Mean Width

Theorem of this section characterizes the convex body with minimalLp-mean width.Theorem 16.
SupposeK∈Kon, p≥1, and T0∈SL(n). Then the following assertions are equivalent:(1)
w-p(K)=wp(T0K).(2)
The measurewp(T0K,·) is isotropic on Sn-1.(3)
For allx∈Rn, the transformation T0 satisfies (50)x2∫Sn-1T0-tu-n+phKpuσdu=n∫Sn-1x,T0-tu2T0-tu-n+p+2hKpuσdu.Proof.
First, we prove the equivalence of (1) and (2). Suppose that (1) holds. Sincew-p(K) is SL(n) invariant, we may assume that T0 is the n×n identity matrix In.
LetT∈L(Rn) be a linear transformation. Then there exists ε0>0 such that, for each ε∈(0,ε0), the matrices In+εT are still positive definite. For ε∈(0,ε0), define (51)Tε=In+εTtIn+εT1/n.Then Tε∈SL(n) is volume preserving. If ε is sufficiently small, from (33), together with the two equalities (see [1]) (52)In+εTu=1+εu,Tu+Oε2,In+εT1/n=1+εntrT+Oε2,and the definition of wp, we have (53)wpTεK=2∫Sn-11+εu,Tu+Oε21+ε/ntrT+Oε2n+phKvσdv.From the smoothness of |Tεtu| in ε, the integrand depends smoothly on ε. Thus, (54)ddεε=0wpTεK=0.Calculating it directly, we have (55)0=2∫Sn-1∂∂εε=01+εu,Tu+Oε21+ε/ntrT+Oε2n+phKvσdv=2n+p∫Sn-1u,Tu-trTnhKvσdv.
Letv∈Sn-1 and T=v⊗v. Using the facts tr(v⊗v)=1 and 〈u,(v⊗v)u〉=〈u,v〉2, it follows that (56)∫Sn-1u,v2dwpK,v=wpKn,where dwp(K,v)=2hKp(v)σ(dv). Thus, wp(K,·) is isotropic on Sn-1.
Next, we show the implication “(2)⇒(1)”. The proof will be completed by two steps.
Firstly, for a pointa=(a1,…,an)∈[0,∞)n, let (57)Fa=2∫Sn-1diaga1,…,anun+phKpvσdv,where diag(a1,…,an) denotes the diagonal matrix with diagonal elements a1,…,an.
We aim to show that(58)Fa≥Fe,whenevera1⋯an=1.Here, e denotes the point (1,…,1).
It can be checked thatF:[0,∞)n→[0,∞) is continuous and convex and F(λa) is strictly increasing in λ∈[0,∞), for any a∈(0,∞)n. Thus, F-1([0,F(e)]) is compact, convex, and of nonempty interior. Namely, it is a convex body.
By using the fact that|diag(a1,…,an)u| is smooth in (a1,…,an) uniformly for u∈Sn-1, we have (59)∂∂aja=eFa=2∫Sn-1∂∂ajdiaga1,…,anun+phKpvσdv=2n+p∫Sn-1uj2dwpK,v.Meanwhile, since the boundary of F-1([0,F(e)]) is given by the equation F(a)=F(e) with a∈R+n, so the vector ∫Sn-1‍(u12,…,un2)dwp(K,v) is an outer normal of F-1([0,F(e)]) at the boundary point e. Notice that wp(K,·) is isotropic; it yields (60)∇Fe=∫Sn-1u12,…,un2dwpK,v=wpK,·ne.Thus, e is an outer normal vector of F-1([0,F(e)]) at the boundary point e. Consequently, (61)F-10,Fe⊂a∈Rn:a,e≤n.That is to say, for all a∈[0,∞)n, if F(a)≤F(e), then 〈a,e〉≤n. In contrast, for all b=(b1,…,bn)∈(0,∞)n with b1⋯bn=1, the arithmetic-geometric mean inequality yields that 〈a,e〉≥n, with equality if and only if b=e. Thus, (58) is derived.
Secondly, with (58) in hand, we aim to show that, for all T∈SL(n), wp(TK)≥wp(K), with equality if and only if T is orthogonal.
It is known thatTt can be represented in the form Tt=PAQ, where P and Q are n×n orthogonal matrices, and A=diag(a1,…,an) is diagonal and positive definite. Therefore, by (33) and (58), we have (62)wpTK=2∫Sn-1Aun+phQKpvσdv=2∫Sn-1diaga1,…,anun+phQKpvσdv≥2∫Sn-1diag1,…,1un+phQKpvσdv=2∫Sn-1hQKpvσdv=wpQK=wpK.Equality holds if and only if (a1,…,an)=(1,…,1), equivalently, if and only if T is orthogonal. Thus, the implication “(2)⇒(1)” is shown.
In the remaining part, we prove the equivalence of (2) and (3). From the definitions ofwp(T0K,·) and (11), we have (63)dwpT0K,u=2hKpT0tuσdu=2T0-tT0tu-n+phKpT0tuσdT0tu,which immediately yields that (64)2∫Sn-1T0-tv-n+phKpvσdv=wpT0K,·,v=T0tu.
Meanwhile, forx∈Rn, we have (65)∫Sn-1x,u2dwpT0K,u=2∫Sn-1x,T0-tT0tu2T0-tT0tu-n+p+2hKpT0tuσdT0tu=2∫Sn-1x,T0-tv2T0-tv-n+p+2hKpvσdv.Therefore, (66)x2∫Sn-1T0-tu-n+phKpuσdu=n∫Sn-1x,T0-tu2T0-tu-n+p+2hKpuσdu.With these, the equivalence of (2) and (3) is shown. The proof is complete.


## 6. The Characterization of the AverageMp(K)

In this section, we consider the average(67)MpK=∫Sn-1xKpσdxof the norm ·K on Sn-1 and define Mp∗(K)=Mp(K∘).We conclude this section with a variation of the minimalLp-mean width position: consider a symmetric convex body K in Rn and the problem of minimizing Mp(TK)Mp∗(TK) over all SL(n).The following fact will be needed.Lemma 17 (see [2]).
LetK be a smooth enough convex body in Rn. We define (68)IpK,θ=∫Sn-1hKp-1u∇hKu,θu,θσdu,θ∈Sn-1.Then (69)wpK2+pIpK,θ=n+p∫Sn-1hKpuu,θ2σdufor every θ∈Sn-1.Now, we obtain the following condition for the minimum position.Theorem 18.
LetK be a symmetric convex body in Rn and assume that Mp(K)Mp∗(K)≤Mp(TK)Mp∗(TK) for every T∈SL(n). Then (70)1MpK∫Sn-1u,θ2uKpσdu=1Mp∗K∫Sn-1u,θ2uK∘pσdu,for every θ∈Sn-1.Proof.
Without loss of generality, we may assume thatK is smooth enough. Let R∈SL(n) and ε>0 be small enough, and write T-1=In+εR. Then (see [1]) (71)Tt=In+εRt-1=In+∑k=1∞-1kεkRkt.Our assumption about K takes the form (72)MpKMp∗K≤∫Sn-1u+εRuKpσdu∫Sn-1u-εRtuK∘pσdu+Oε2.Note that there are two equalities (see [33]) (73)u+εRuKp=hK∘u+εRup=hK∘up+εphK∘up-1∇hK∘u,Ru+Oε2,u-εRtuK∘p=hKu-εRtup=hKup-εphKup-1∇hKu,Rtu+Oε2,which implies (74)MpKMp∗K≤MpKMp∗K-pMpK∫Sn-1εhKp-1u∇hKu,Rtuσdu+pMp∗K∫Sn-1εhK∘p-1u∇hK∘u,Ruσdu+Oε2.Letting ε→0+ and replacing R by -R, we have (75)1MpK∫Sn-1phK∘p-1u∇hK∘u,Ruσdu=1Mp∗K∫Sn-1phKp-1u∇hKu,Rtuσdufor every R∈SL(n). Using (75) with Rθ(x)=〈x,θ〉θ, θ∈Sn-1, we get (76)1MpK∫Sn-1hK∘p-1u∇hK∘u,θu,θσdu=1Mp∗K∫Sn-1hKp-1u∇hKu,θu,θσdufor every θ∈Sn-1. Taking into account Lemma 17, we conclude the proof.Whenp=1, it is easy to show that Theorem 18 reduces to Giannopoulos and Milman’s result (see [1]).Recall that the conditiona(TK)=1 means that TK⊆B but there exist contact points of TK and B (see [1]). We also obtain the following result.Theorem 19.
LetK be a symmetric convex body in Rn satisfying a(K)=1 and Mp(K)≤Mp(TK) for every T∈GL(n) with a(TK)=1. Then, for every θ∈Sn-1 we can find contact points x1, x2 of K and B such that (77)1+px1,θ2≤n+pMpK∫Sn-1uKpu,θ2σdu≤1+px2,θ2.Proof.
LetT∈Ln(Rn) and ε>0 be small enough. Then (78)T1≔minu∈Sn-1u+εTuKIn+εT-1satisfies a(T1K)=1. Therefore (79)∫Sn-1u+εTupσdu≥MpKminx∈Sn-1x+εTxp.Notice that (see [33]) (80)u+εTuKp=uKp+εpuKp-1∇hK∘u,Tu+Oε2.It follows from (79) that (81)∫Sn-1puKp-1∇hK∘u,Tuσdu+Oε≥MpKminx∈Sn-1x+εTxp-1ε.
Letxε be a point on Sn-1 at which the minimum is attained. Since xε∈Sn-1 and ·K≥|·|, and noticing that (82)xε+εTxεp=1+εpxε,Txε+Oε2,(81) takes the form (83)∫Sn-1puKp-1∇hK∘u,Tuσdu+Oε≥MpKminx∈Sn-1x+εTxp-1ε≥MpKxε+εTxεp-1ε=MpKpxε,Txε+Oε.If x is a contact point of K and B, we must have (84)1+εT≥x+εTxK≥xε+εTxεK≥xεK-εT.It follows that (85)1≤xεK≤1+2εT.Also, x∈Sn-1 and using (85) we get x=limk→∞xεk=1. That is, x is a contact point of K and B. Now, we can find a sequence εk→0 and a point x∈Sn-1 such that xεk→x. Letting k→∞ in (83), we obtain (86)∫Sn-1uKp-1∇hK∘u,Tuσdu≥MpKx,Tx.
ReplacingT by -T we find another contact point x′ of K and B such that (87)∫Sn-1uKp-1∇hK∘u,Tuσdu≤MpKx′,Tx′.Choosing Tθ(x)=〈x,θ〉θ, θ∈Sn-1, and applying Lemma 17, we obtain (77).Whenp=1, it is easy to show that Theorem 19 reduces to Giannopoulos and Milman’s result (see [1]).



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