Origin-symmetric Convex Bodies Research Articles

On the sine polarity and the Lp-sine Blaschke-Santaló inequality

This paper is dedicated to study the sine version of polar bodies and establish the Lp-sine Blaschke-Santaló inequality for the Lp-sine centroid body.The Lp-sine centroid body ΛpK for a star body K⊂Rn is a convex body based on the Lp-sine transform, and its associated Blaschke-Santaló inequality provides an upper bound for the volume of Λp∘K, the polar body of ΛpK, in terms of the volume of K. Thus, this inequality can be viewed as the “sine cousin” of the Lp Blaschke-Santaló inequality established by Lutwak and Zhang. As p→∞, the limit of Λp∘K becomes the sine polar body K⋄ and hence the Lp-sine Blaschke-Santaló inequality reduces to the sine Blaschke-Santaló inequality for the sine polar body. The sine polarity naturally leads to a new class of convex bodies Cen, which consists of all origin-symmetric convex bodies generated by the intersection of origin-symmetric closed solid cylinders. Many notions in Cen are developed, including the cylindrical support function, the supporting cylinder, the cylindrical Gauss image, and the cylindrical hull. Based on these newly introduced notions, the equality conditions of the sine Blaschke-Santaló inequality are settled.

Local 𝐿^{𝑝}-Brunn–Minkowski inequalities for 𝑝<1

The L p L^p -Brunn–Minkowski theory for p ≥ 1 p\geq 1 , proposed by Firey and developed by Lutwak in the 90’s, replaces the Minkowski addition of convex sets by its L p L^p counterpart, in which the support functions are added in L p L^p -norm. Recently, Böröczky, Lutwak, Yang and Zhang have proposed to extend this theory further to encompass the range p ∈ [ 0 , 1 ) p \in [0,1) . In particular, they conjectured an L p L^p -Brunn–Minkowski inequality for origin-symmetric convex bodies in that range, which constitutes a strengthening of the classical Brunn-Minkowski inequality. Our main result confirms this conjecture locally for all (smooth) origin-symmetric convex bodies in R n \mathbb {R}^n and p ∈ [ 1 − c n 3 / 2 , 1 ) p \in [1 - \frac {c}{n^{3/2}},1) . In addition, we confirm the local log-Brunn–Minkowski conjecture (the case p = 0 p=0 ) for small-enough C 2 C^2 -perturbations of the unit-ball of ℓ q n \ell _q^n for q ≥ 2 q \geq 2 , when the dimension n n is sufficiently large, as well as for the cube, which we show is the conjectural extremal case. For unit-balls of ℓ q n \ell _q^n with q ∈ [ 1 , 2 ) q \in [1,2) , we confirm an analogous result for p = c ∈ ( 0 , 1 ) p=c \in (0,1) , a universal constant. It turns out that the local version of these conjectures is equivalent to a minimization problem for a spectral-gap parameter associated with a certain differential operator, introduced by Hilbert (under different normalization) in his proof of the Brunn–Minkowski inequality. As applications, we obtain local uniqueness results in the even L p L^p -Minkowski problem, as well as improved stability estimates in the Brunn–Minkowski and anisotropic isoperimetric inequalities.

The measure-comparison problem for polar (p,μ)-centroid bodies

For p>−1 and α-homogeneous measure μ on Rn with α≠−p, this paper considers the measure-comparison problem for the polar (p,μ)-centroid operator Γp,μ⁎. Namely, if K,L are origin-symmetric convex bodies in Rn with satisfyingΓp,μ⁎K⊂Γp,μ⁎L, does it follow thatpα+pμ(L)p≤pα+pμ(K)p? This paper shows that the answer is confirmative when −1<p<0,α+p<0, and negative when p≥0, or −1<p<0,α+p>0. In the negative cases, we also give a solution to the isomorphic version of the above problem. Namely, the condition Γp,μ⁎K⊂Γp,μ⁎L implies thatμ(L)1/α≤nμ(K)1/α.

The radii of sections of origin-symmetric convex bodies and their applications

Let V and W be any convex and origin-symmetric bodies in Rn . Assume that for some A ∈LRn→Rn, detA≠0, V is contained in the ellipsoid A−1B2n, where B2n is the unit Euclidean ball. We give a lower bound for the W-radius of sections of A−1V in terms of the spectral radius of A∗A and the expectations of ‖⋅‖V and ‖⋅‖Wo with respect to Haar measure on Sn−1⊂Rn. It is shown that the respective expectations are bounded as n→∞ in many important cases. As an application we offer a new method of evaluation of n-widths of multiplier operators. As an example we establish sharp orders of n-widths of multiplier operators Λ:LpMd→LqMd, 1<q≤2≤p<∞ on compact homogeneous Riemannian manifolds Md. Also, we apply these results to prove the existence of flat polynomials on Md.

Diagonal Minkowski classes, zonoid equivalence, and stable laws

We consider the family of convex bodies obtained from an origin symmetric convex body [Formula: see text] by multiplication with diagonal matrices, by forming Minkowski sums of the transformed sets, and by taking limits in the Hausdorff metric. Support functions of these convex bodies arise by an integral transform of measures on the family of diagonal matrices, equivalently, on Euclidean space, which we call [Formula: see text]-transform. In the special case, if [Formula: see text] is a segment not lying on any coordinate hyperplane, one obtains the family of zonoids and the cosine transform. In this case two facts are known: the vector space generated by support functions of zonoids is dense in the family of support functions of origin symmetric convex bodies; and the cosine transform is injective. We show that these two properties are equivalent for general [Formula: see text]. For [Formula: see text] being a generalized zonoid, we determine conditions that ensure the injectivity of the [Formula: see text]-transform. Relations to mixed volumes and to a geometric description of one-sided stable laws are discussed. The later probabilistic application motivates our study of a family of convex bodies obtained as limits of sums of diagonally scaled [Formula: see text]-balls.

General volumes in the Orlicz–Brunn–Minkowski theory and a related Minkowski problem II

The general dual volume $\dveV(K)$ and the general dual Orlicz curvature measure $\deV(K, \cdot)$ were recently introduced for functions $G: (0, \infty)\times \sphere\rightarrow (0, \infty)$ and convex bodies $K$ in $\R^n$ containing the origin in their interiors. We extend $\dveV(K)$ and $\deV(K, \cdot)$ to more general functions $G: [0, \infty)\times \sphere\rightarrow [0, \infty)$ and to compact convex sets $K$ containing the origin (but not necessarily in their interiors). Some basic properties of the general dual volume and of the dual Orlicz curvature measure, such as the continuous dependence on the underlying set, are provided. These are required to study a Minkowski-type problem for the dual Orlicz curvature measure. We mainly focus on the case when $G$ and $\psi$ are both increasing, thus complementing our previous work. The Minkowski problem asks to characterize Borel measures $\mu$ on $\sphere$ for which there is a convex body $K$ in $\R^n$ containing the origin such that $\mu$ equals $\deV(K, \cdot)$, up to a constant. A major step in the analysis concerns discrete measures $\mu$, for which we prove the existence of convex polytopes containing the origin in their interiors solving the Minkowski problem. For general (not necessarily discrete) measures $\mu$, we use an approximation argument. This approach is also applied to the case where $G$ is decreasing and $\psi$ is increasing, and hence augments our previous work. When the measures $\mu$ are even, solutions that are origin-symmetric convex bodies are also provided under some mild conditions on $G$ and $\psi$. Our results generalize several previous works and provide more precise information about the solutions of the Minkowski problem when $\mu$ is discrete or even.

On the Comparison of Measures of Convex Bodies via Projections and Sections

Abstract We study the inequalities between measures of convex bodies implied by comparison of their projections and sections. Recently, Giannopoulos and Koldobsky proved that if $K, L$ are convex bodies such that the volume of the projection of $K$ onto any hyperplane through the origin is bounded by the volume of the intersection of $L$ with the hyperplane, then $|K| \le |L|$. Firstly, we study the reverse question: in particular, we show that if $K, L$ are origin-symmetric convex bodies in John’s position such that the volume of the intersection of $K$ with any hyperplane through the origin is bounded by the volume of the projection of $L$ onto the hyperplane, then $|K| \le \sqrt {n}|L|$. The condition we consider is weaker than the conditions that appear in the Busemann–Petty and Shephard problems. Secondly, we appropriately extend the result of Giannopoulos and Koldobsky to various classes of measures possessing concavity properties, including log-concave measures.

The dual Minkowski problem for symmetric convex bodies

The dual Minkowski problem for even data asks what are the necessary and sufficient conditions on a prescribed even measure on the unit sphere for it to be the q-th dual curvature measure of an origin-symmetric convex body in Rn. A full solution to this is given when 1<q<n. The necessary and sufficient conditions turn out to be an explicit measure concentration condition. To obtain the results, a variational approach is used, where the functional is the sum of a dual quermassintegral and an entropy integral. The proof requires two crucial estimates. The first is an estimate of the entropy integral which is obtained by using a spherical partition. The second is a sharp estimate of the dual quermassintegrals for a carefully chosen barrier convex body.

The Reverse Petty Projection Inequality

It is proved that if K is an origin-symmetric convex body in R2 and Π*K is the polar projection body of K, then the volumes of K and Π*K satisfy the inequality V(K)V(Π*K) ⩾ 2 with equality if K is a parallelogram.

The log-Brunn-Minkowski inequality in ℝ³

Böröczky, Lutwak, Yang, and Zhang have recently proved the log-Brunn-Minkowski inequality for two origin-symmetric convex bodies in the plane which is stronger than the classical Brunn-Minkowski inequality. This paper establishes the log-Brunn-Minkowski, log-Minkowski, L p L_p -Minkowski, and L p L_p -Brunn-Minkowski inequalities for two classes of convex bodies in R 3 \mathbb {R}^3 .

Continuity of the solution to the even logarithmic Minkowski problem in the plane

This paper investigates continuity of the solution to the even logarithmic Minkowski problem in the plane. It is shown that the weak convergence of a sequence of cone-volume measures in ℝ2 implies the convergence of the sequence of the corresponding origin-symmetric convex bodies in the Hausdorff metric.

Equality Case in van der Corput’s Inequality and Collisions in Multiple Lattice Tilings

Van der Corput’s provides the sharp bound $$\mathop {\mathrm {vol}}\nolimits (C) \le m 2^d$$ on the volume of a d-dimensional origin-symmetric convex body C that has $$2m-1$$ points of the integer lattice in its interior. For $$m=1$$ , a characterization of the equality case $$\mathop {\mathrm {vol}}\nolimits (C)= m 2^d$$ is equivalent to the well-known problem of characterizing tilings by translations of a convex body. It is rather surprising that so far, for $$m \ge 2$$ , no characterization of the equality case has been available, though a hint to the respective characterization problem can be found in the 1987 monograph of Gruber and Lekkerkerker. We give an explicit characterization of the equality case for all $$m \ge 2$$ . Our result reveals that, the equality case for $$m \ge 2$$ is more restrictive than for $$m=1$$ . We also present consequences of our characterization in the context of multiple lattice tilings.

Log-Minkowski inequalities for the L_{p}-mixed quermassintegrals

Böröczky et al. proposed the log-Minkowski problem and established the plane log-Minkowski inequality for origin-symmetric convex bodies. Recently, Stancu proved the log-Minkowski inequality for mixed volumes; Wang, Xu, and Zhou gave the L_{p} version of Stancu’s results. In this paper, we define the L_{p}-mixed quermassintegrals probability measure and obtain the log-Minkowski inequality for the L_{p}-mixed quermassintegrals. As its application, we establish the L_{p}-mixed affine isoperimetric inequality. In addition, we also consider the dual log-Minkowski inequalities for the L_{p}-dual mixed quermassintegrals.

A sharp dual $$L_{p}$$ L p John ellipsoid problem for $$p\le -n-1$$ p ≤ - n - 1

When $$p>0$$ , the dual $$L_p$$ John ellipsoids provide a unified treatment for the Lowner ellipsoid and the Legendre ellipsoid associated with a convex body. When $$p<0$$ , very little relevance to known ellipsoids has been found for the dual $$L_p$$ John ellipsoid so far. In this paper we investigate a sharp dual $$L_p$$ John ellipsoid problem associated with origin-symmetric convex bodies when $$p\le -n-1$$ . The solution unifies the classic John ellipsoid and the classic Petty ellipsoid.

Generalized dual Sudakov minoration via dimension-reduction—a program

We propose a program for establishing a conjectural extension to the class of (origin-symmetric) log-concave probability measures $\mu$, of the classical dual Sudakov Minoration on the expectation of the supremum of a Gaussian process: \begin{equation} \label{eq:abstract} M(Z_p(\mu), C \int ||x||_K d\mu \cdot K) \leq \exp(C p) \;\;\, \forall p \geq 1 . \end{equation} Here $K$ is an origin-symmetric convex body, $Z_p(\mu)$ is the $L_p$-centroid body associated to $\mu$, $M(A,B)$ is the packing-number of $B$ in $A$, and $C > 0$ is a universal constant. The Program consists of first establishing a Weak Generalized Dual Sudakov Minoration, involving the dimension $n$ of the ambient space, which is then self-improved to a dimension-free estimate after applying a dimension-reduction step. The latter step may be thought of as a conjectural small-ball one-sided variant of the Johnson--Lindenstrauss dimension-reduction lemma. We establish the Weak Generalized Dual Sudakov Minoration for a variety of log-concave probability measures and convex bodies (for instance, this step is fully resolved assuming a positive answer to the Slicing Problem). The Separation Dimension-Reduction step is fully established for ellipsoids and, up to logarithmic factors in the dimension, for cubes, resulting in a corresponding Generalized (regular) Dual Sudakov Minoration estimate for these bodies and arbitrary log-concave measures, which are shown to be (essentially) best-possible. Along the way, we establish a regular version of (\ref{eq:abstract}) for all $p \geq n$ and provide a new direct proof of Sudakov Minoration via The Program.

A Flag Representation for a n-Dimensional Convex Body

The cosine representation of the support function of a centrally symmetric convex body plays a fundamental role in integral geometry. In this article, one new so-called flag representation for the support function of an origin symmetric n-dimensional convex body in terms of surface curvature functions of the convex body is found. Using the representation, we propose a sufficient condition for an origin symmetric n-dimensional convex body to be a zonoid. The condition has a local equatorial description.

Necessary subspace concentration conditions for the even dual Minkowski problem

We prove tight subspace concentration inequalities for the dual curvature measures C˜q(K,⋅) of an n-dimensional origin-symmetric convex body for q≥n+1. This supplements former results obtained in the range q≤n.

Superconcentration, and randomized Dvoretzky's theorem for spaces with 1-unconditional bases

Let n be a sufficiently large natural number and let B be an origin-symmetric convex body in Rnin the ℓ-position, and such that the space (Rn,‖⋅‖B) admits a 1-unconditional basis. Then for any ε∈(0,1/2], and for random cεlog⁡n/log⁡1ε-dimensional subspace E distributed according to the rotation-invariant (Haar) measure, the section B∩E is (1+ε)-Euclidean with probability close to one. This shows that the “worst-case” dependence on ε in the randomized Dvoretzky theorem in the ℓ-position is significantly better than in John's position. It is a previously unexplored feature, which has strong connections with the concept of superconcentration introduced by S. Chatterjee. In fact, our main result follows from the next theorem: Let B be as before and assume additionally that B has a smooth boundary and Eγn‖⋅‖B≤ncEγn‖gradB(⋅)‖2 for a small universal constant c>0, where gradB(⋅) is the gradient of ‖⋅‖B and γn is the standard Gaussian measure in Rn. Then for any p∈[1,clog⁡n] the p-th power of the norm ‖⋅‖Bp is Clog⁡n-superconcentrated in the Gauss space.

Volume inequalities of convex bodies from cosine transforms on Grassmann manifolds

The Lp cosine transform on Grassmann manifolds naturally induces finite dimensional Banach norms whose unit balls are origin-symmetric convex bodies in Rn. Reverse isoperimetric type volume inequalities for these bodies are established, which extend results from the sphere to Grassmann manifolds.

Thick sections of convex bodies

Let P and Q be origin-symmetric convex bodies in Rn such that for every slab (thick section) of a fixed width 2t, symmetric about the origin, its intersections with P and Q have equal volumes. Is then necessarily P=Q? We show that this is true in the class of origin-symmetric convex polytopes. We also study a modified version of this problem, when one of the bodies is a Euclidean ball and both bodies have the same volume. Finally we discuss a related problem concerning sections of convex bodies by hyperplanes that are distance t from the origin.