Erwin Lutwak, Deane Yang, Gaoyong Zhang
A direct approach to Ball's simplex inequality is presented. This approach, which does not use the Brascamp-Lieb inequality, also gives Barthe's characterization of the simplex for Ball's inequality and extends it from discrete to arbitrary measures. It also yields the dual inequality, along with equality conditions, and it does both for arbitrary measures.
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In this paper, we establish mean width inequalities of sections and projections of convex bodies for isotropic measures with complete equality conditions, which extends the recent work of Alonso-Gutiérrez and Brazitikos. Different from their approach, our proof is based on the approach developed by Lutwak, Yang and Zhang, by using the Ball-Barthe inequality, the mass transportation, and the isotropic embedding.
Sharp isoperimetric inequalities for the sine transform of even isotropic measures are established. The corresponding reverse inequalities are obtained in an asymptotically optimal form. These new inequalities have direct applications to strong volume estimates for convex bodies from data about their sections or projections.
We prove a scale of generalized $L^p$-Poincaré inequalities and Sobolev type inequalities on graphs with polynomial volume growth. They are optimal on Vicsek graphs.
In this paper, we deal with the asymmetric Orlicz zonotopes by using the method of shadow system. We establish the volume product inequality and volume ratio inequality for asymmetric Orlicz zonotopes, along with their equality cases.
Extemizers in Soprunov and Zvavitch's Bezout inequalities for mixed volumes
本文研究了凸体满足Soprunov和Zvavitch提出的混合体积Bezout不等式的边界结构必要条件，并给出了三维情形的新解。
Strengthened volume inequalities for L_p zonoids of even isotropic measures
本文加强了偶对称各向同性测度的L_pzonoids的体积不等式，同时证明了Brascamp-Lieb不等式的加强版本，并给出了反向等周不等式的稳定性版本。
This paper contains a number of results related to volumes of projective perturbations of convex bodies and the Laplace transform on convex cones. First, it is shown that a sharp version of Bourgain's slicing conjecture implies the Mahler conjecture for convex bodies that are not necessarily centrally-symmetric. Second, we find that by slightly translating the polar of a centered convex body, we may obtain another body with a bounded isotropic constant. Third, we provide a counter-example to a conjecture by Kuperberg on the distribution of volume in a body and in its polar.
Inequalities and counterexamples for functional intrinsic volumes and beyond
本文研究了函数内在体积的不等式失效问题，通过反例和变分函数量建立了Wulff型不等式，并推广了Alexandrov-Fenchel不等式，恢复了Polya-Szegő型不等式的一个特例。
Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining additive maps of star bodies. These inequalities provide generalizations of results for projection and intersection bodies. As a corollary, a new Brunn-Minkowski inequality is obtained for the volume of polar projection bodies.
We prove several inequalities estimating the distance between volumes of two bodies in terms of the maximal or minimal difference between areas of sections or projections of these bodies. We also provide extensions in which volume is replaced by an arbitrary measure.
A note on Bézout type inequalities for mixed volumes and Minkowski sums
本文研究了R n中凸体混合体积与Minkowski和的Bézout型不等式，通过质量传输方法推广了Jian Xiao的混合判别式不等式，并推导出混合体积及Minkowski和的相应不等式。
We obtain optimal inequalities for the volume of the polar of random sets, generated for instance by the convex hull of independent random vectors in Euclidean space. Extremizers are given by random vectors uniformly distributed in Euclidean balls. This provides a random extension of the Blaschke-Santalo inequality which, in turn, can be derived by the law of large numbers. The method involves generalized shadow systems, their connection to Busemann type inequalities, and how they interact with functional rearrangement inequalities.
We develop a technique using dual mixed-volumes to study the isotropic constants of some classes of spaces. In particular, we recover, strengthen and generalize results of Ball and Junge concerning the isotropic constants of subspaces and quotients of L_p and related spaces. An extension of these results to negative values of p is also obtained, using generalized intersection-bodies. In particular, we show that the isotropic constant of a convex body which is contained in an intersection-body is bounded (up to a constant) by the ratio between the latter's mean-radius and the former's volume-ra…
The classical Loomis-Whitney inequality and the uniform cover inequality of Bollobás and Thomason provide lower bounds for the volume of a compact set in terms of its lower dimensional coordinate projections. We provide further extensions of these inequalities in the setting of convex bodies. We also establish the corresponding dual inequalities for coordinate sections; these uniform cover inequalities for sections may be viewed as extensions of Meyer's dual Loomis-Whitney inequality.
The theory of dual mixed volumes is extended to star bodies in cotangent bundles and is used to prove several isosystolic inequalities for Hamiltonian systems and Finsler metrics.
We consider a generalization of the hyperplane problem to arbitrary measures in place of volume and to sections of lower dimensions. We prove this generalization for unconditional convex bodies and for duals of bodies with bounded volume ratio. We also prove it for arbitrary symmetric convex bodies under the condition that the dimension of sections is less than $λn$ for some $λ\in (0,1).$ The constant depends only on $λ.$ Finally, we show that the behavior of the minimal sections for some measures may be different from the case of volume.
Radon Transforms of Projection Functions and Aleksandrov-Fenchel Inequalities
本文针对平均截面体的对数凹性问题，建立新的亚历山德罗夫-芬切尔不等式，证明平均截面体的所有内在体积均为对数凹的。
We show that the hyperplane conjecture holds for the classes of $k$-intersection bodies with arbitrary measures in place of volume.
Mixed volumes, which are the polarization of volume with respect to the Minkowski addition, are fundamental objects in convexity. In this note we announce the construction of mixed integrals, which are functional analogs of mixed volumes. We build a natural addition operation + on the class of quasi-concave functions, such that every class of α-concave functions is closed under +. We then define the mixed integrals, which are the polarization of the integral with respect to +. We proceed to discuss the extension of various classic inequalities to the functional setting. For general quasi-conca…
On isoperimetric local-Bollobás-Thomason inequalities
该研究证明了关于凸体的等周型不等式，通过构造合适的Hanner多面体来保证投影体积的下界。
This paper develops basic setting for the dual Orlicz-Brunn-Minkowski theory for star bodies. An Orlicz $φ$-radial addition of two or more star bodies is proposed and related dual Orlicz-Brunn-Minkowski inequality is established. Based on a linear Orlicz $φ$-radial addition of two star bodies, we derive a formula for the Orlicz $L_ϕ$-dual mixed volume. Moreover, a dual Orlicz-Minkowski inequality for the Orlicz $L_ϕ$-dual mixed volume, a dual Orlicz isoperimetric inequality for the Orlicz $L_ϕ$-dual surface area and a dual Orlicz-Urysohn inequality for the Orlicz $L_ϕ$-harmonic mean radius are…
A matroid polytope approach to sharp affine isoperimetric inequalities for volume decomposition functionals
本文通过拟阵多面体方法，证明体分解泛函的递推公式，建立其尖锐仿射等周不等式，并展示拟阵理论在凸几何中的应用。
This article belongs to the area of geometric tomography, which is the study of geometric properties of solids based on data about their sections and projections. We describe a new direction in geometric tomography where different volumetric results are considered in a more general setting, with volume replaced by an arbitrary measure. Surprisingly, such a general approach works for a number of volumetric results. In particular, we discuss the Busemann-Petty problem on sections of convex bodies for arbitrary measures and the slicing problem for arbitrary measures. We present generalizations of…
We obtain a new extension of Rogers-Shephard inequality providing an upper bound for the volume of the sum of two convex bodies $K$ and $L$. We also give lower bounds for the volume of the $k$-th limiting convolution body of two convex bodies $K$ and $L$. Special attention is paid to the $(n-1)$-th limiting convolution body, for which a sharp inequality, which is equality only when $K=-L$ is a simplex, is given. Since the $n$-th limiting convolution body of $K$ and $-K$ is the polar projection body of $K$, these inequalities can be viewed as an extension of Zhang's inequality.
We prove the validity of the $p$-Brunn-Minkowski inequality for the intrinsic volume $V_k$, $k=2,\dots, n-1$, of convex bodies in $\mathbb{R}^n$, in a neighborhood of the unit ball, for $0\le p<1$. We also prove that this inequality does not hold true on the entire class of convex bodies of $\mathbb{R}^n$, when $p$ is sufficiently close to $0$.
We study the class of (locally) anti-blocking bodies as well as some associated classes of convex bodies. For these bodies, we prove geometric inequalities regarding volumes and mixed volumes, including Godberson's conjecture, near-optimal bounds on Mahler volumes, Saint-Raymond-type inequalities on mixed volumes, and reverse Kleitman inequalities for mixed volumes. We apply our results to the combinatorics of posets and prove Sidorenko-type inequalities for linear extensions of pairs of 2-dimensional posets. The results rely on some elegant decompositions of differences of anti-blocking bodie…
We consider convex sets whose modulus of convexity is uniformly quadratic. First, we observe several interesting relations between different positions of such ``2-convex'' bodies; in particular, the isotropic position is a finite volume-ratio position for these bodies. Second, we prove that high dimensional 2-convex bodies posses one-dimensional marginals that are approximately Gaussian. Third, we improve for 1<p<=2 some bounds on the isotropic constant of quotients of subspaces of L_p and S_p^m, the Schatten Class space.
An Improved Volume Ratio Bound via Isotropic Positions
该研究针对凸体的体积比界问题，通过将凸体置于各向同性位置并结合相关估计与新的无维界，改进了已沿用近二十五年的通用体积比上界。
In this paper we prove different functional inequalities extending the classical Rogers-Shephard inequalities for convex bodies. The original inequalities provide an optimal relation between the volume of a convex body and the volume of several symmetrizations of the body, such as, its difference body. We characterize the equality cases in all these inequalities. Our method is based on the extension of the notion of a convolution body of two convex sets to any pair of log-concave functions and the study of some geometrical properties of these new sets.
Sharp Small-Volume Isoperimetry from Log-Sobolev Inequalities
本文通过对数索博列夫不等式，建立了满足CD(0,∞)的加权黎曼流形乘积的渐近等周剖面在小体积下的尖锐极限恒等式，并推广到非光滑子空间，同时得到大偏差与中偏差渐近。
A theorem of W. Derrick ensures that the volume of any Riemannian cube $([0,1]^n,g)$ is bounded below by the product of the distances between opposite codimension-1 faces. In this paper, we establish a discrete analog of Derrick's inequality for weighted open covers of the cube $[0,1]^n$, which is motivated by a question about lower volume bounds in metric spaces. Our main theorem generalizes a previous result of the author, which gave a combinatorial version of Derrick's inequality and was used in the analysis of boundaries of hyperbolic groups. As an application, we answer a question of Y. B…
The classical Dvoretzky--Rogers lemma provides a deterministic algorithm by which, from any set of isotropic vectors in Euclidean $d$-space, one can select a subset of $d$ vectors whose determinant is not too small. Subsequently, Pelczyński and Szarek improved this lower bound by a factor depending on the dimension and the number of vectors. Pivovarov, on the other hand, determined the expectation of the square of the volume of parallelotopes spanned by $d$ independent random vectors in $\mathbb{R}^d$, each one chosen according to an isotropic measure. We extend Pivovarov's result to a class o…
In this paper, we give a systolic inequality for a quotient space of a Carnot group $Γ\backslash G$ with Popp's volume. Namely we show the existence of a positive constant $C$ such that the systole of $Γ\backslash G$ is less than ${\rm Cvol}(Γ\backslash G)^{\frac{1}{Q}}$, where $Q$ is the Hausdorff dimension. Moreover, the constant depends only on the dimension of the grading of the Lie algebra $\mathfrak{g}=\bigoplus V_i$. To prove this fact, the scalar product on $G$ introduced in the definition of Popp's volume plays a key role.
Several years ago the authors started looking at some problems of convex geometry from a more general point of view, replacing volume by an arbitrary measure. This approach led to new general properties of the Radon transform on convex bodies including an extension of the Busemann-Petty problem and a slicing inequality for arbitrary functions. The latter means that the sup-norm of the Radon transform of any probability density on a convex body of volume one is bounded from below by a positive constant depending only on the dimension. In this note, we prove an inequality that serves as an umbre…
On the volume of sums of anti-blocking bodies
研究反阻塞体Minkowski和体积的不等式，证明了Plünnecke-Ruzsa型不等式、V.Milman不等式及Roger-Shephard不等式的反演性，给出了精确常数。
Volume and Projection Inequalities I: Zonoids and Courtade's Conjecture
该论文研究带状体的体积与投影不等式，证明对数次模性猜想等价于zonotope体积多项式的瑞利性质，在次数/余次数≤3时成立，次数/余次数≥4时构造反例，还否定了科塔德猜想在三维及以上带状体中的成立性。
Various results are proved giving lower bounds for the $m$th intrinsic volume $V_m(K)$, $m=1,\dots,n-1$, of a compact convex set $K$ in ${\mathbb{R}}^n$, in terms of the $m$th intrinsic volumes of its projections on the coordinate hyperplanes (or its intersections with the coordinate hyperplanes). The bounds are sharp when $m=1$ and $m=n-1$. These are reverse (or dual, respectively) forms of the Loomis-Whitney inequality and versions of it that apply to intrinsic volumes. For the intrinsic volume $V_1(K)$, which corresponds to mean width, the inequality obtained confirms a conjecture of Betke …
In the course of classifying generic sparse polynomial systems which are solvable in radicals, Esterov recently showed that the volume of the Minkowski sum $P_1+\dots+P_d$ of $d$-dimensional lattice polytopes is bounded from above by a function of order $O(m^{2^d})$, where $m$ is the mixed volume of the tuple $(P_1,\dots,P_d)$. This is a consequence of the well-known Aleksandrov-Fenchel inequality. Esterov also posed the problem of determining a sharper bound. We show how additional relations between mixed volumes can be employed to improve the bound to $O(m^d)$, which is asymptotically sharp.…
Lutwak's volume inequalities for polar projection bodies of all orders are generalized to polarizations of Minkowski valuations generated by even, zonal measures on the Euclidean unit sphere. This is based on analogues of mixed projection bodies for such Minkowski valuations and a generalization of the notion of centroid bodies. A new integral representation is used to single out Lutwak's inequalities as the strongest among these families of inequalities, which in turn are related to a conjecture on affine quermassintegrals. In the dual setting, a generalization of volume inequalities for inte…
We prove several stability and volume difference inequalities for projections of convex bodies and apply them to prove a hyperplane inequality for surface area of projection bodies.
On the Log-submodularity for zonoids: from Mixed Volume inequalities to the Hypercube
该研究证明了四维带状体的对数次模型不等式，扩展了三维相关结果，提出任意维度带状体的该类不等式猜想，归约后揭示了带状体混合体积、拟阵理论与实代数几何的关联。
We prove a randomized version of the generalized Urysohn inequality relating mean-width to the other intrinsic volumes. To do this, we introduce a stochastic approximation procedure that sees each convex body K as the limit of intersections of Euclidean balls of large radii and centered at randomly chosen points. The proof depends on a new isoperimetric inequality for the intrinsic volumes of such intersections. If the centers are i.i.d. and sampled according to a bounded continuous distribution, then the extremizing measure is uniform on a Euclidean ball. If one additionally assumes that the …
Motivated by conjectures of Mahler and Makai Jr., we study bounds on the volume of a convex body in terms of the successive minima of its polar body.
Let z_0,...,z_n be the (n-1)-dimensional volumes of facets of an n-simplex. Then we have the simplex inequalities: z_p < z_0+...+\check{z}_p+...+z_n (0 =< p =< n), generalizations of triangle inequalities. Conversely, suppose that numbers z_0,...,z_n > 0 satisfy these inequalities. Does there exist an n-simplex the volumes of whose facets are them? Kakeya solved this problem affirmatively in the case n = 3 and conjectured that the assertion is affirmative also for all n >= 4. We prove that his conjecture is affirmative. To do this, we define three kinds of spaces of loops associated to n-simpl…
For an isotropic convex body $K\subset\mathbb{R}^n$ we consider the isotropic constant $L_{K_N}$ of the symmetric random polytope $K_N$ generated by $N$ independent random points which are distributed according to the cone probability measure on the boundary of $K$. We show that with overwhelming probability $L_{K_N}\leq C\sqrt{\log(2N/n)}$, where $C\in(0,\infty)$ is an absolute constant. If $K$ is unconditional we argue that even $L_{K_N}\leq C$ with overwhelming probability. The proofs are based on concentration inequalities for sums of sub-exponential or sub-Gaussian random variables, respe…
Tightening Inequalities on Volume Extremal $k$-Ellipsoids Using Asymmetry Measures
本文利用 John 不对称性统一并加强了关于低维椭球体积上下界的两类不等式，证明其在大维度、平面及任意不对称性凸体（含交叉多胞形、平行多胞形和单纯形）中的紧性，并给出与 Banach-Mazur 距离相关的推论。
Mixed volumes and the Bochner method
本文利用谱定理将 Alexandrov-Fenchel 不等式及其矩阵对应物简化为平凡的 Bochner 公式，提供了概念与技术上更简单的新证明并澄清了底层结构。
Let $1\leq i \leq k < n$ be integers. We prove the following exact inequalities for any convex body $K\subset\mathbb{R}^n$ with centroid at the origin, and any $k$-dimensional subspace $E\subset \mathbb{R}^n$: \begin{align*} &V_i \big( K\cap E \big) \geq \left( \frac{i+1}{n+1} \right)^i \max_{x\in K} V_i \big( ( K-x) \cap E \big) , \\ &\widetilde{V}_i \big( K\cap E \big) \geq \left( \frac{i+1}{n+1} \right)^i \max_{x\in K} \widetilde{V}_i \big( ( K-x) \cap E \big) ; \end{align*} $V_i$ is the $i$th intrinsic volume, and $\widetilde{V}_i$ is the $i$th dual volume taken within $E$. Our results are…
On polynomial inequalities for cone-volumes of polytopes
受离散对数Minkowski问题启发，研究给定矩阵U的锥体积集，证明其为路径连通的半代数集，并定义子空间浓度多胞体，为离散对数Minkowski问题提供新几何视角。
In this paper we consider the problem of minimizing the relative perimeter under a volume constraint in the interior of a convex body, i.e., a compact convex set in Euclidean space with interior points. We shall not impose any regularity assumption on the boundary of the convex set. Amongst other results, we shall prove the equivalence between Hausdorff and Lipschitz convergence, the continuity of the isoperimetric profile with respect to the Hausdorff distance,and the convergence in Hausdorff distance of sequences of isoperimetric regions and their free boundaries. We shall also describe the …
Equality conditions for the fractional superadditive volume inequalities
本文针对 Lebesgue 测度的分数超加性体积不等式，证明了一维情形下基于区间条件的等式成立条件，以及 $n\geq2$ 维情形下集合和测度为零的等式条件。
Consider a proper geodesic metric space $(X,d)$ equipped with a Borel measure $μ.$ We establish a family of uniform Poincaré inequalities on $(X,d,μ)$ if it satisfies a local Poincaré inequality ($P_{loc}$) and a condition on growth of volume. Consequently if $μ$ is doubling and supports $(P_{loc})$ then it satisfies a $(σ,β,σ)$-Poincaré inequality. If $(X,d,μ)$ is a $δ$-hyperbolic space then using the volume comparison theorem in \cite{BCS} we obtain a uniform Poincaré inequality with exponential growth of the Poincaré constant. If $X$ is the universal cover of a compact $CD(K,\infty)$ space …
We prove that on an essentially non-branching $\mathrm{MCP}(K,N)$ space, if a geodesic ball has a volume lower bound and satisfies some additional geometric conditions, then in a smaller geodesic ball (in a quantified sense) we have an estimate on the isoperimetric constants.
Mixed volumes of zonoids and the absolute value of the Grassmannian
该研究提出组合框架，将带形域混合体积与格拉斯曼流形绝对值研究相联。利用多面体计算，在(n,d)=(6,2)和(6,3)时得出 d 维中 n 个带形域新不等式族，其具有在各带形域中闵可夫斯基线性的特殊特征。
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