A New Affine Invariant Geometric Functional for Polytopes and Its Associated Affine Isoperimetric Inequalities

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Affine Isoperimetric Inequalities,”

International Mathematics Research Notices, Vol. 2021, No. 12, pp. 8977–8995 Advance Access Publication May 15, 2019

https://doi.org/10.1093/imrn/rnz090

A New Affine Invariant Geometric Functional for Polytopes and Its Associated Affine Isoperimetric Inequalities

Jiaqi Hu

¹

and Ge Xiong

^2,∗

1

School of Mathematical Sciences, Fudan University, Shanghai 200433, PR China and

²

School of Mathematical Sciences, Tongji University, Shanghai, 200092, PR China

∗Correspondence to be sent to: e-mail: xiongge@tongji.edu.cn

A new affine invariant geometric functional for convex polytopes is introduced. Some new sharp affine isoperimetric inequalities are established for this new functional, which are extensions of Lutwak–Yang–Zhang’s results on their celebrated cone-volume functional.

1 Introduction

A convex body (i.e., a compact convex subset with nonempty interior) K in n- dimensional Euclidean space Rⁿ is uniquely determined by its support function h_K : Sⁿ⁻¹ → R, which is defined for u ∈ Sⁿ⁻¹ byh_K(u) = max{u·x : x ∈ K}, where Sⁿ⁻¹ is the unit sphere andu·x denotes the standard inner product ofu andx. The projection body K of K is defined as the convex body whose support function, for u∈Sⁿ⁻¹, is given byh_K(u)=vol_n₋₁(K|u^⊥), where vol_n₋₁denotes(n−1)-dimensional volume andK|u^⊥denotes the image of orthogonal projection ofKonto the codimension 1 subspace orthogonal tou. The support function ofKcan also be represented as

h_K(u)=1 2

Sⁿ⁻¹

|u·v|dS_K(v), u∈Sⁿ⁻¹, (1.1)

whereS_K is thesurface area measureof convex bodyK. Formula (1.1) follows from the Cauchy projection formula. See, for example, [28, page 569] for details.

Communicated by Prof. Igor Rivin

Received March 14, 2018; Revised October 16, 2018; Accepted April 9, 2019

For permissions, please e-mail: journals.permission@oup.com.

Downloaded from https://academic.oup.com/imrn/article/2021/12/8977/5487946 by Tongji University user on 28 June 2021

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The projection body is one of the most important objects in convex geometry, and has been intensively investigated during the past three decades. See, for example, [1], [6], [7], [14], [16], [17], [33], etc. It is centro-affine invariant, that is, forT ∈SL(n),(TK)= T⁻^t(K), whereT⁻^t denotes the inverse of the transpose of T. It is worth mentioning that there stands a celebrated unsolved problem regarding projection bodies, called Schneider’s projection problem: asKranges over the class of origin-symmetric convex bodies inRⁿ, what is the least upper bound of the affine-invariant ratio

V(K)/V(K)ⁿ⁻¹¹_n ,

whereV(K)denotes then-dimensional volume ofK. See, for example, [27], [28], and [29].

The lower bound for the affine-invariant ratio is also unknown; Petty [24] conjectured that the minimum of this quantity is attained precisely by ellipsoids.

An effective tool to study Schneider’s projection problem is the cone-volume functional U, which was introduced by Lutwak, Yang, and Zhang (LYZ) [19]: if P is a convex polytope inRⁿthat contains the originoin its interior, thenU(P)is defined as

U(P)ⁿ= 1 nⁿ

ui1∧···∧uin=0

h_i

1· · ·h_i

na_i

1· · ·a_i

n, (1.2)

where u₁,. . .,u_N are the outer normal unit vectors to the faces of P, h₁,. . .,h_N are the corresponding distances of the faces from the origin and a₁,. . .,a_N are the corresponding areas of the faces.

It is interesting that the functionalUis centro-affine invariant, that is,U(TP)= U(P), forT∈SL(n). LetV_i=a_ih_i/n, i=1,. . .,N, then

U(P)ⁿ=

ui1∧···∧u_in=0

V_i₁· · ·V_i_n.

SinceV(P)ⁿ =(_N

i=1V_i)ⁿ, it follows that U(P) < V(P). IfP is a random polytope with a large number of faces,U(P)is very close toV(P). See LYZ [19, page 1772] for details. It is this important property of the functionalUthat makes it so powerful.

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For instance, using the functionalU, LYZ [19] presented an affirmative answer to the modified Schneider projection problem: if P is an origin-symmetric polytope in Rⁿ, then

V(P)

U(P)ⁿ²V(P)ⁿ²⁻¹ ≤2ⁿ nⁿ

n!

¹₂

, (1.3)

with equality if and only ifPis a parallelotope. This statement is especially interesting as for an origin-symmetric convex bodyK, it is known thatV(K)/V(K)ⁿ⁻¹is not maxi- mized by parallelotopes on one hand and (1.3) yields an upper bound onV(K)/V(K)ⁿ⁻¹ of optimal order on the other hand. See, for example, Schneider [28, page 578] for details.

LYZ [19] also proved that for an origin-symmetric convex polytope P in Rⁿ, it holds

V(^∗P)U(P)ⁿ²V(P)ⁿ²⁻¹≥ 2ⁿ (nⁿn!)¹²

, (1.4)

with equality if and only ifPis a parallelotope. Here,^∗Pis the polar body ofP.

Similarly, for a convex polytopePinRⁿwith its John point at the origin, LYZ [19]

established

V(P)

U(P)ⁿ²V(P)ⁿ²⁻¹ ≤ nⁿ(n+1)ⁿ⁺²¹ (n!)³²

, (1.5)

with equality if and only ifP is a simplex. Recall that the John point of a convex body is precisely the center of its John ellipsoid, which is the ellipsoid contained in the body with maximal volume. See, for example, [13] and [21].

Showing the lower bound of functional U in terms of volume V makes an interesting story. LYZ [19] conjectured that for polytopesP with centroid at the origin, there holds

U(P)≥ (n!)ⁿ¹

n V(P), (1.6)

with equality if and only ifPis a parallelotope.

It took more than a dozen years to completely settle this conjecture. In [10], He, Leng, and Li proved (1.6) for origin-symmetric polytopes, including its equality condition. In [31], the 2nd author of this article gave a simplified proof for symmetric polytopes and proved (1.6), including the equality case, for 2D and 3D polytopes with centroid at the origin. A complete solution to this conjecture was attributed to Henk and Linke [11].

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In 2015, Bor¨ oczky and LYZ [4] extended the domain of cone-volume functional¨ U to the class of convex bodiesKinRⁿwith the origin in their interiors and defined

U(K)ⁿ= 1 nⁿ

u1∧···∧un=0h_K(u₁)· · ·h_K(u_n)dS_K(u₁)· · ·dS_K(u_n). (1.7)

SinceV(K)ⁿ=(_n¹ _Sn−1h_KdS_K)ⁿ, it follows thatU(K)≤V(K).U(K)is still centro- affine invariant, that is,U(TK)=U(K), forT ∈ SL(n). Recently, B¨or¨oczky and Henk [2]

proved that LYZ’s conjecture is also affirmative for convex bodies with centroid at the origin.

In view of itsvolume attribute of the cone-volume functionalU, together with its strong applications, the main goal of this article is to further generalize the cone- volume functionalU(K)to the so-calledmixed cone-volume functional U₁(K,L)as

U₁(K,L)ⁿ=

u1∧···∧un=0

dV_K,L(u₁)· · ·dV_K,L(u_n),

where V_K,L(·)is the newly introduced mixed cone-volume measure. See Definition3.1 and Definition3.2for details.

It is striking that as the important geometric functional 1st mixed volume V₁(K,L) (see Section2 for its definition) generalizes the volume functional V(K), the mixed cone-volume functionalU₁(K,L)not only generalizes the cone-volume functional U(K) but also has very similar properties to V₁(K,L). However, what we want to emphasize here is thatU₁(K,L)andV₁(K,L)have different features. As an illustration, we can see later that if K and L are polytopes, then U₁(K,L) < V₁(K,L). Thus, U₁ is indeed a new geometric functional for polytopes.

In this article, several sharp affine isoperimetric inequalities forU₁(K,L) are established.

Theorem 1.1. IfP,Qare convex polytopes inRⁿandQis origin-symmetric, then

V(P)V(Q)

U₁(P,Q)ⁿ²V₁(P,Q)ⁿ² ≤2ⁿ nⁿ

n!

¹

2

, (1.8)

with equality if and only ifPandQare parallel parallelotopes.

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Theorem 1.2. IfP,Qare convex polytopes inRⁿandQis origin-symmetric, then

V(^∗P)U₁(P,Q)ⁿ²V₁(P,Q)ⁿ² ≥ 2ⁿ (nⁿn!)¹²

V(Q), (1.9)

Theorem 1.3. If P,Q are convex polytopes in Rⁿ and the John point of Q is at the origin, then

V(P)V(Q)

U₁(P,Q)ⁿ²V₁(P,Q)ⁿ² ≤ nⁿ(n+1)ⁿ⁺²¹ (n!)³²

, (1.10)

with equality if and only ifPandQare parallel simplices.

It is clear that ifQ =P in the above theorems, then the inequalities (1.8), (1.9), and (1.10) precisely turn to the inequalities (1.3), (1.4), and (1.5), respectively.

This paper is organized as follows. After listing some basic facts on convex bodies in Section2, we introduce the notion ofmixed cone-volume measure V_K,L, as well as themixed cone-volume functional U₁(K,L), of convex bodies KandL in Section3.

Then some fundamental properties of V_K,L andU₁(K,L)are established. The proofs of Theorem1.1, Theorem1.2, and Theorem1.3are provided in Section4.

2 Preliminaries

For quick later reference, we collect some basic facts on convex bodies. Excellent references are the books by Gardner [5], Gruber [9], Schneider [28], and Thompson [30].

WriteKⁿandK_oⁿfor the set of convex bodies and the set of convex bodies with the origin in their interior in Rⁿ, respectively. LetPⁿ ⊆Kⁿdenote the class of convex polytopes. The standard unit ball ofRⁿis denoted byB= {x ∈Rⁿ:|x| ≤1}. Its volume isω_n=π^n/2/ (1+n/2). Forx∈Rⁿ\ {o}, letx = |x|⁻¹x.

LetK ∈ Kⁿ. By the definition of support function, it follows that forλ ≥0 and T∈GL(n),

h_K(λx)=λh_K(x), h_TK(x)=h_K(T^tx), x∈Rⁿ. (2.1)

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Theradial functionρ_K ofK∈Kⁿ_o is defined by

ρ_K(x)=max{λ≥0 :λx∈K}, x∈Rⁿ\ {o}.

Forλ >0 andT∈GL(n), it yields that

ρ_K(λx)=λ⁻¹ρ_K(x), ρ_TK(x)=ρ_K(T⁻¹x), x∈Rⁿ\ {o}. (2.2)

Thepolar body K^∗ofK∈Kⁿ_o is defined by

K^∗= {x∈Rⁿ: x·y≤1, y∈K}.

ForK∈KⁿandT∈GL(n), we have

ρ_K∗ =h⁻_K¹ and (TK)^∗=T^−tK^∗. (2.3)

Thesurface area measure S_KofK∈Kⁿis a finite Borel measure onSⁿ⁻¹, defined for the Borel setω⊆Sⁿ⁻¹by

S_K(ω)=Hⁿ⁻¹(ν_K⁻¹(ω)),

where ν_K : ∂K → Sⁿ⁻¹ is the Gauss map of K, defined on ∂K, the set of points of

∂K that have a unique outer unit normal. Recall that the Gauss mapν_K exists almost everywhere on ∂K with respect to the (n−1)-dimensional Hausdorff measure Hⁿ⁻¹, that is,Hⁿ⁻¹(∂K\∂K)=0. Thus, for any continuousf :Sⁿ⁻¹→R, it holds

Sⁿ⁻¹

f(u)dS_K(u)=

∂K

f(ν_K(x))dHⁿ⁻¹(x). (2.4)

The cone-volume measure V_K of K ∈ Kⁿ is a finite Borel measure on Sⁿ⁻¹, defined for the Borel setω⊆Sⁿ⁻¹by

V_K(ω)= 1 n

ω

h_K(u)dS_K(u). (2.5)

Observe thatV_K is SL(n)-invariant, that is,

V_TK(ω)=V_K(T^tω), forT ∈SL(n), (2.6)

whereT^tω = {T^tu:u∈ω}.

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TheMinkowski combinationofK,L∈Kⁿis defined by

λK+μL= {λx+μy x∈K, y∈L}, λ,μ≥0.

The1st mixed volume V₁(K,L)of convex bodiesK,Lis defined by

V₁(K,L)= 1 n lim

→0⁺

V(K+L)−V(K)

= 1

n

Sⁿ⁻¹

h_L(u)dS_K(u). (2.7)

From the affine invariance of volume, it follows that V₁(TK,TL) = |detT|V₁(K,L), for T∈GL(n). Thus, for any continuousf :Sⁿ⁻¹→R, we have

Sⁿ⁻¹

f(u)dS_TK(u)= |detT|

Sⁿ⁻¹

f(T⁻^tu)|T⁻^tu|dS_K(u). (2.8)

See, for example, Schneider [28] for its proof.

For convenience, we introduce the U-functional for measures. Let μbe a finite Borel measure onSⁿ⁻¹. Then theU-functional U(μ)ofμis defined by

U(μ)ⁿ=

u1∧···∧un=0

dμ(u₁)· · ·dμ(u_n). (2.9)

U(μ)is centro-affine invariant in the sense that

U(Tμ)=U(μ), T∈SL(n), (2.10)

whereTμis the affine image of measureμ, that is,

Tμ(ω)=μ(T⁻¹ω), for Borel setω⊆Sⁿ⁻¹. (2.11)

A finite positive Borel measureμonSⁿ⁻¹is said to beisotropicif

Sⁿ⁻¹|u·v|²dμ(v)=1, for allu∈Sⁿ⁻¹. Specially, the discrete measureμis isotropic if

v∈suppμ

|u·v|²μ({v})=1, for allu∈Sⁿ⁻¹.

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Forx₁,. . .,x_n∈Rⁿ, [o,x₁]+ · · · +[o,x_n] is a parallelotope. Write [x₁,. . .,x_n] for its n-dimension volume. Supposeμis an isotropic measure onSⁿ⁻¹. Then

1 n!

Sⁿ⁻¹

· · ·

Sⁿ⁻¹

[u₁,. . .,u_n]²dμ(u₁)· · ·dμ(u_n)=1. (2.12)

See, for example, LYZ [20] or [22] for its proof.

SupposeZis a convex body inRⁿwith its support function

h_Z(u)=

Sⁿ⁻¹|u·v|dμ(v), u∈Sⁿ⁻¹,

whereμis a finite Borel measure onSⁿ⁻¹. The McMullen–Matheron–Weil formula reads

V(Z)= 2ⁿ n!

Sⁿ⁻¹

· · ·

Sⁿ⁻¹

[u₁,. . .,u_n]dμ(u₁)· · ·dμ(u_n). (2.13)

See, e.g., Matheron [23] or Weil [31] for its proof.

3 The Mixed Cone-Volume Functional and the Mixed LYZ Ellipsoid

In this section, we introduce a new notion: the mixed cone-volume functionalU₁(K,L).

To define this functional, a new measure, called the mixed cone-volume measureV_K,L, is involved. It is pointed out that the mixed cone-volume measure V_K,L is a natural extension of the important cone-volume measureV_K.

Definition 3.1. Let K,L ∈ Kⁿ and L contains the origin. The mixed cone-volume measure V_K,LofKandLis defined by

V_K,L(ω)= 1 n

ω

h_L(u)dS_K(u), for Borel setω⊆Sⁿ⁻¹.

Note that the total mass ofV_K,Lis exactly the 1st mixed volumeV₁(K,L), that is, V_K,L(Sⁿ⁻¹) =V₁(K,L). In additional, ifLis the unit ballB, thenV_K,B = S_K/n; ifL = K, then V_K,K = V_K. This means that the mixed cone-volume measure V_K,L contains two fundamental measures in geometry: the surface area measureS_K and the cone volume measure V_K. It is well known that the surface area measure is characterized by the classicalMinkowski problem, which is one of the cornerstones of the Brunn–Minkowski theory of convex bodies. In recent years, cone-volume measures have appeared and were studied in various contexts, see, for example, [2], [3], [4], [10], [11], and [32].

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IfP∈PⁿandQ∈Kⁿwitho∈Q, thenV_P,Qcan be represented as

V_P,Q(ω)= 1 n

u∈suppSP

h_Q(u)S_P(u)δ_u(ω), for Borel setω⊆Sⁿ⁻¹, (3.1)

whereδ_u(·)is the Delta measure onSⁿ⁻¹concentrated onu.

Definition 3.2. Let K,L ∈ Kⁿ and L contains the origin. The mixed cone-volume functional U₁(K,L)ofKandL, is defined by

U₁(K,L)ⁿ=

u1∧···∧un=0dV_K,L(u₁)· · ·dV_K,L(u_n).

Observe that ifL =K, it immediately yields U₁(K,L)=U₁(K,K)=U(K), which turns to theU functional introduced by Bor¨¨ oczky and LYZ [4]. By (2.9), it follows that U₁(K,L)=U(V_K,L).

Specifically, ifP ∈PⁿandQ∈Kⁿwitho∈Q, thenU₁(P,Q)can be represented as

U₁(P,Q)ⁿ= 1 nⁿ

u1∧···∧un=0

h_Q(u₁)· · ·h_Q(u_n)S_P(u₁)· · ·S_P(u_n), (3.2)

whereu_i∈suppS_P, i=1. . .,n. Moreover, ifQ=P, thenU₁(P,Q)=U(P), which goes to LYZ’s original definition (1.2).

Several observations for the relations betweenU₁(K,L)andV₁(K,L)are in order.

First, for general convex bodies, it always holdsU₁(K,L)≤V₁(K,L), which can be seen from the fact thatV₁(K,L)ⁿ=( _Sn−1dV_K,L(u))ⁿ. Second, if convex bodyKis smooth, then U₁(K,L) = V₁(K,L). See Böröczky and LYZ [4] for details. Third, ifK is a polytope, in light of

V₁(K,L)ⁿ= 1 nⁿ

u1,...,un∈suppSK

h_L(u₁)· · ·h_L(u_n)S_K(u₁)· · ·S_K(u_n),

it follows thatU₁(K,L) < V₁(K,L). Since U₁(K,L)is affine invariant by Proposition3.6, U₁is indeed a new affine invariant geometric functional for polytopes.

To prove the main results, we need to use the mixed LYZ ellipsoid introduced by Hu–Xiong–Zou. For more information on the mixed LYZ ellipsoid and its associated affine isoperimetric inequalities, refer to [12].

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Definition 3.3. LetK,L∈ KⁿandLcontains the origin in its interior. Themixed LYZ ellipsoid₋₂(K,L)ofKandLis defined by

ρ⁻²

−2(K,L)(u)= n V₁(K,L)

Sⁿ⁻¹

|u·v| h_L(v)

₂

dV_K,L(v), foru∈Sⁿ⁻¹.

It is clear that when L = K, then ₋₂(K,K) = ₋₂K, which is precisely the celebrated LYZ ellipsoids defined in [18]. The LYZ ellipsoid is in some sense dual to the classical Legendre ellipsoid of inertia in mechanics. When viewed as suitably normalized matrix-valued operators on the space of convex bodies, it was proved by Ludwig [15] that the Legendre ellipsoid and the LYZ ellipsoid are the only linearly invariant operators that satisfy the inclusion–exclusion principle.

LetM =(m_ij)_n×nbe the symmetric positive definite matrix with entries

m_ij= 1 V₁(K,L)

Sⁿ⁻¹

(v·e_i)(v·e_j)h⁻¹_L (v)dS_K(v),

wheree_i. . .,e_nare the orthonormal basis forRⁿ. Then

₋₂(K,L)= {x∈Rⁿ x·Mx≤1}.

IfP ∈PⁿandQ∈Kⁿwith the origin in its interior, then₋₂(P,Q)is defined by

ρ⁻²

−2(P,Q)(u)= 1 V₁(K,L)

v∈suppSP

|u·v|²S_P(v)

h_Q(v), foru∈Sⁿ⁻¹. (3.3)

Recall that ν_K : ∂K → Sⁿ⁻¹ is the Gauss map of K, defined on ∂K, the set of points of∂Kthat have a unique outer unit normal.

Proposition 3.4. Suppose thatK∈KⁿandT ∈GL(n). Forx∈∂K, then

ν_TK(y)= T⁻^tν_K(x), fory=Tx∈∂(TK).

Proof. By the definition of support function, it follows thath_K(ν_K(x))=x·ν_K(x), for x∈∂K. According to (2.1), we have

h_TK(T⁻^tν_K(x))=h_K(ν_K(x))=x·ν_K(x)=Tx·T⁻^tν_K(x).

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Thus,

h_TK(T^−tν_K(x))=y· T^−tν_K(x).

Fromy=Tx∈∂(TK), it follows thatν_TK(y)= T⁻^tν_K(x). Proposition 3.5. The mixed cone-volume measureV_K,L is SL(n)-invariant. That is, for T∈SL(n)and Borelω⊆Sⁿ⁻¹, thenV_TK,TL(ω)=V_K,L(T^tω).

Proof. Let x ∈ ∂K. Then y = Tx ∈ ∂(TK). Since T ∈ SL(n), from (2.6) and Proposition3.4, it follows that

dV_TK(ν_TK(y))=dV_K(T^tν_TK(y))=dV_K(ν_K(x)).

Thus,

1

ny·ν_TK(y)dHⁿ⁻¹(y)= 1

nx·ν_K(x)dHⁿ⁻¹(x).

By Proposition3.4, we have

dHⁿ⁻¹(y)= |T^−tν_K(x)|dHⁿ⁻¹(x). (3.4)

Meanwhile, if Tx = y ∈ ν_TK⁻¹(ω), then ν_TK(y) ∈ ω. By Proposition 3.4, it follows that

ν_K(x)∈ T^tω,

which is equivalent tox∈ν_K⁻¹(T^tω). Thus,

ν_TK⁻¹(ω)=T(ν_K⁻¹(T^tω)). (3.5)

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From Definition 3.1, (2.4), (3.5), Proposition 3.4, (3.4), (2.4), and Definition 3.1 again, it follows that

V_TK,TL(ω)= 1 n

ω

h_TL(u)dS_TK(u)

= 1 n

ν_TK⁻¹(ω)

h_TL(ν_TK(y))dHⁿ⁻¹(y)

= 1 n

T(ν_K⁻¹(T^tω))

h_TL(ν_TK(y))dHⁿ⁻¹(y)

= 1 n

ν_K⁻¹(T^tω)

h_TL(T^−tν_K(x))|T^−tν_K(x)|dHⁿ⁻¹(x)

= 1 n

ν_K⁻¹(T^tω)

h_L(ν_K(x))dHⁿ⁻¹(x)

= 1 n

T^tωh_L(u)dS_K(u)

=V_K,L(T^tω).

This completes the proof.

Proposition 3.6. The mixed cone-volume functional U₁(K,L)is SL(n)-invariant. That is, forT∈SL(n), thenU₁(TK,TL)=U₁(K,L).

Proof. From Proposition3.5and (2.11), it follows that for Borelω⊆Sⁿ⁻¹,

V_TK,TL(ω)=V_K,L(T^tω)=T⁻^tV_K,L(ω). (3.6)

So, from Definition3.2combining with (2.9), (3.6), and (2.10), it follows that

U₁(TK,TL)=U(V_TK,TL)=U(T^−tV_K,L)=U(V_K,L)=U₁(K,L).

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Proposition 3.7. The mixed LYZ ellipsoid ₋₂(K,L) is affine invariant. That is, for T∈GL(n), then₋₂(TK,TL)=T(₋₂(K,L)).

Proof. Foru∈Sⁿ⁻¹, from Definition3.3, (2.8), (2.1), and (2.2), it follows that

ρ⁻²

−2(TK,TL)(u)

= n

V₁(TK,TL)

Sⁿ⁻¹

|u·v| h_TL(v)

₂

dV_TK,TL(v)

= |detT|

|detT|V₁(K,L)

sⁿ⁻¹

|u· T⁻^tv|²h⁻¹_TL(T⁻^tv)|T⁻^tv|dS_K(v)

= 1

V₁(K,L)

Sⁿ⁻¹|u·T^−tv|²h⁻_TL¹(T^−tv)dS_K(v)

= 1

V₁(K,L)

Sⁿ⁻¹|T⁻¹u·v|²h⁻_L¹(v)dS_K(v)

= n

V₁(K,L)

Sⁿ⁻¹

|T⁻¹u·v| h_L(v)

2

dV_K,L(v)

=ρ⁻²

−2(K,L)(T⁻¹u)=ρ⁻²_T(

−2(K,L))(u).

4 Affine Isoperimetric Inequalities for the Mixed Cone-Volume Functional

The following inequality relates the mixed cone-volume functional U₁(K,L) with the volume of the mixed LYZ ellipsoid₋₂(K,L).

Theorem 4.1. Suppose that P ∈ Pⁿ and Q ∈ Kⁿ with the origin in its interior.

Then

V(₋₂(P,Q))V(P) U₁(P,Q)ⁿ²V₁(P,Q)ⁿ² ≤

nⁿ n!

¹

2

ω_n,

with equality if and only if [u₁,. . .,u_n]/h_Q(u₁)· · ·h_Q(u_n)is independent of the choice of u₁,. . .,u_non suppS_P, wheneveru₁∧ · · · ∧u_n=0.

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Proof. LetP∈Pⁿ. From (1.1), the support function ofPis given by

h_P(u)=1 2

v∈suppSP

|u·v|S_P(v), u∈Sⁿ⁻¹.

According to the McMullen–Matheron–Weil formula (2.13), we have

V(P)= 1 n!

u1,...,un∈suppSP

[u₁,. . .,u_n]S_P(u₁)· · ·S_P(u_n). (4.1)

From (1.2), Propositions3.6and 3.7, it follows that the left side of the desired inequality is SL(n)-invariant. In order to prove the theorem we may assume, w.l.o.g.,

₋₂(P,Q)=

V(₋₂(P,Q)) ω_n

¹_n B.

Then from (3.3), foru∈Sⁿ⁻¹, it follows that

ω_n V(₋₂(P,Q))

²_n

= 1

V₁(P,Q)

v∈suppSP

|u·v|²S_P(v) h_Q(v),

which implies the discrete measure

μ:= 1 V₁(P,Q)

V(₋₂(P,Q)) ω_n

²

n S_P h_Q

is isotropic onSⁿ⁻¹. Thus, by (2.12), we have

ω_n V(₋₂(P,Q))

₂

= 1

n!V₁(P,Q)ⁿ

u1,...,un∈suppSP

[u₁,. . .,u_n]²S_P(u₁)· · ·S_P(u_n)

h_Q(u₁)· · ·h_Q(u_n). (4.2)

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From (4.2), the Jensen inequality and (4.1), it follows that ω_n

V(₋₂(P,Q))

2 n!V₁(P,Q)ⁿ nⁿU₁(P,Q)ⁿ

= 1

nⁿU₁(P,Q)ⁿ

u1,...,un∈suppSP

[u₁,. . .,u_n] h_Q(u₁)· · ·h_Q(u_n)

2

h_Q(u₁)· · ·h_Q(u_n)S_P(u₁)· · ·S_P(u_n)

= 1

nⁿU₁(P,Q)ⁿ

u1∧···∧un=0

[u₁,. . .,u_n] h_Q(u₁)· · ·h_Q(u_n)

2

h_Q(u₁)· · ·h_Q(u_n)S_P(u₁)· · ·S_P(u_n)

≥

⎛

⎝ 1 nⁿU₁(P,Q)ⁿ

u1∧···∧un=0

[u₁,. . .,u_n]

h_Q(u₁)· · ·h_Q(u_n)h_Q(u₁)· · ·h_Q(u_n)S_P(u₁)· · ·S_P(u_n)

⎞

⎠

2

=

⎛

⎝ 1 nⁿU₁(P,Q)ⁿ

u1,...,un∈suppSP

[u₁,. . .,u_n]S_P(u₁)· · ·S_P(u_n)

⎞

⎠

2

=

n!V(P) nⁿU₁(P,Q)ⁿ

2

,

with equality if and only if

[u₁,. . .,u_n] h_Q(u₁)· · ·h_Q(u_n)

is independent of the choice ofu₁,. . .,u_non suppS_P, wheneveru₁∧ · · · ∧u_n=0.

To prove the main results, the following volume ratio inequality for₋₂(K,L), which was previously established by Hu–Xiong–Zou [12], is needed.

Lemma 4.2. SupposeP,Q∈Pⁿ.

(1). IfQis origin symmetric, then

V(₋₂(P,Q))≥ ω_n 2ⁿV(Q),

(2). If the John point ofQis at the origin, then

V(₋₂(P,Q))≥ n!ω_n nⁿ²(n+1)ⁿ⁺²¹

V(Q),

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Theorem 4.3. Suppose thatP,Q∈PⁿandQis origin symmetric. Then

V(P)V(Q)

U₁(P,Q)ⁿ²V₁(P,Q)ⁿ² ≤2ⁿ nⁿ

n!

¹₂ ,

Proof. From Theorem4.1and Lemma4.2(1), we have

ω_n 2ⁿ

V(P)V(Q)

U₁(P,Q)ⁿ²V₁(P,Q)ⁿ² ≤ V(₋₂(P,Q))V(P) U₁(P,Q)ⁿ²V₁(P,Q)ⁿ² ≤

nⁿ n!

¹₂ ω_n,

which yields the desired inequality.

If the equality holds in the inequality, by the equality condition of Lemma4.2 (1), it follows thatPandQare parallel parallelotopes.

Conversely, assumeQ=[−1, 1]ⁿ, then the polar bodyQ^∗is a cross-polytope with vertices{uρ_Q∗(u):u∈suppS_Q}. Since suppS_Q=suppS_P, by the identity

[u₁,. . .,u_n]

h_Q(u₁)· · ·h_Q(u_n)=[u₁ρ_Q_∗(u₁),. . .,u_nρ_Q_∗(u_n)],

it follows that it is constant for allu₁,. . .,u_n on suppS_P, whenever u₁∧ · · · ∧u_n = 0.

By the equality conditions of Theorem4.1and Lemma4.2(1), the equality holds. This

completes the proof.

Recall that the Reisner inequality reads: IfKis a projection body inRⁿ, then

V(K)V(K^∗)≥ 4ⁿ n!,

with equality if and only ifKis a parallelotope. See, for example, Reisner [25,26] and Gordon–Meyer–Reisner [8].

Theorem 4.4. Suppose thatP,Q∈PⁿandQis origin symmetric. Then

V(^∗P)U₁(P,Q)ⁿ²V₁(P,Q)ⁿ² ≥ 2ⁿ (nⁿn!)¹²

V(Q),

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Proof. From Theorem4.3and Reisner’s inequality, it follows that

4ⁿ n!

V(Q)

V(^∗P)U₁(P,Q)ⁿ²V₁(P,Q)ⁿ² ≤ V(P)V(Q)

U₁(P,Q)ⁿ²V₁(P,Q)ⁿ² ≤2ⁿ nⁿ

n!

¹₂ ,

IfP andQ are parallel parallelotopes, thenP is still a parallelotope. By the equality conditions of Theorem 4.3 and Reisner’s inequality, the equality holds. This

Theorem 4.5. Suppose thatP,Q∈Pⁿand the John point ofQis at the origin. Then

V(P)V(Q)

U₁(P,Q)ⁿ²V₁(P,Q)ⁿ² ≤ nⁿ(n+1)ⁿ⁺²¹ (n!)³²

,

Proof. From Theorem4.1and Lemma4.2(2), we have

n!ω_n nⁿ²(n+1)ⁿ⁺²¹

V(P)V(Q)

U₁(P,Q)ⁿ²V₁(P,Q)ⁿ² ≤ V(₋₂(P,Q))V(P) U₁(P,Q)ⁿ²V₁(P,Q)ⁿ² ≤

nⁿ n!

¹₂ ω_n,

If the equality holds in the inequality, by the equality condition of Lemma4.2 (2), it follows thatPandQare parallel simplices.

Conversely, assumeQis a regular simplex inRⁿ, then the polar bodyQ^∗is also a regular simplex with vertices{uρ_Q∗(u):u∈ suppS_Q}. Since suppS_Q =suppS_P, by the identity

[u₁,. . .,u_n]

h_Q(u₁)· · ·h_Q(u_n) =[u₁ρ_Q_∗(u₁),. . .,u_nρ_Q_∗(u_n)],

it follows that it is constant for allu₁,. . .,u_n on suppS_P, wheneveru₁∧ · · · ∧u_n = 0.

By the equality conditions of Theorem 4.1and Lemma 4.2(2), the equality holds. This

One obvious question regarding the functionalsU₁andV₁beg to be asked.

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Problem 4.6. SupposeKandLare convex bodies inRⁿ. Is there an absolute constant conly depending on the dimensionn, such that

U₁(K,L) V₁(K,L) ≥c?

Funding

This work was supported by NSFC [No. 11871373].

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