Keith Ball
The largest discs contained in a regular tetrahedron lie in its faces. The proof is closely related to the theorem of Fritz John characterising ellipsoids of maximal volume contained in convex bodies.
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A convex body K has associated with it a unique circumscribed ellipsoid CE(K) with minimum volume, and a unique inscribed ellipsoid IE(K) with maximum volume. We first give a unified, modern exposition of the basic theory of these extremal ellipsoids using the semi-infinite programming approach pioneered by Fritz John in his seminal 1948 paper. We then investigate the automorphism groups of convex bodies and their extremal ellipsoids. We show that if the automorphism group of a convex body K is large enough, then it is possible to determine the extremal ellipsoids CE(K) and IE(K) exactly, usin…
Using previous results about shadow systems and Steiner symmetrization, we prove that the local maximizers of the volume product of convex bodies are actually the global maximizers, that is: ellipsoids.
This paper is devoted to measures of symmetry based on distance between centroid and one of the centers of John and Lowner ellipsoid. The author proves the accuracy of the derived upper bounds for the considered measures of symmetry.
{We show in this paper that two normal elliptic sections through every point of the boundary of a smooth convex body essentially characterize an ellipsoid and furthermore, that four different pairwise non-tangent elliptic sections through every point of the $C^2$-differentiable boundary of a convex body also essentially characterize an ellipsoid.
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。
Strengthened inequalities for the mean width and the $\ell$-norm of origin symmetric convex bodies
本文改进了对称凸体均宽和ℓ-范数的极值结果，证明了立方体和正交叉多面体在特定条件下的最优稳定性版本，并探讨了相关稳定性结果。
Several characterizations of complex ellipsoids among convex bodies in Cn, in terms of their sections and projections are proved. Characterizing complex symmetry in similar terms is an important tool.
Regular ellipsoids and a Blaschke-Santaló-type inequality for projections of non-symmetric convex bodies
本文将 Pisier 关于覆盖数正则递减的定理推广至非对称凸体，通过引入关于 Santaló 点位于原点的凸体投影的 Blaschke-Santaló 型不等式作为关键新方法，并探讨了其在各向同性凸体平均范数中的应用。
Strengthened inequalities for the mean width and the $\ell$-norm
本文加强了关于凸体均宽和ℓ-范数的不等式，证明了正则单纯形在特定条件下的极值性质，并研究了中心各向同性测度支撑集的凸包的相关稳定性结果。
We investigate a duality relation between floating and illumination bodies. The definitions of these two bodies suggest that the polar of the floating body should be similar to the illumination body of the polar. We consider this question for the class of centrally symmetric convex bodies. We provide precise estimates for $(B_p^n)$ and for centrally symmetric convex bodies with everywhere positive Gauss curvature. Our estimates show that equality of the polar of the floating body and the illumination body of the polar can only be achieved in the case of ellipsoids.
General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the Minkowski sum boundary of $m$ arbitrary ellipsoids in $N$-dimensional Euclidean space. Expressions for the principal curvatures of these Minkowski sums are also derived. These results are then used to obtain upper and lower volume bounds for the Minkowski sum of ellipsoids in terms of their defining matrices; the lower bounds are sharper than the Brunn-Minkowski inequality. A reverse isometric inequality for convex bodies is also given.
Sections and projections of nested convex bodies
本文研究了嵌套凸体的截面与投影，通过分析两凸体的截面或投影关系，揭示了球体和椭球体的新描述方法。
The purpose of this paper is to study the reflections of a convex body. In particular, we are interested in orthogonal reflections of its sections that can be extended to reflections of the whole body. For this reason, we need to study the case of a non-spherical ellipsoid, where a surprising structure arises (Section 2). These results allow us to give a new characterization of ellipsoids in terms of their reflections and, on the other hand, to prove a result deeply related to a conjecture due to K. Bezdek.
We give a variety of uniqueness results for minimal ellipsoids circumscribing and maximal ellipsoids inscribed into a convex body. Uniqueness follows from a convexity or concavity criterion on the function used to measure the size of the ellipsoid. Simple examples with non-unique minimal or maximal ellipsoids conclude this article.
Abstract In this paper, we study affine bodies of revolution. This will allow us to prove that a convex body all whose orthogonal $n$-projections are affinely equivalent is an ellipsoid, provided $n\equiv 0,1, 2$ mod $4$, $n>1$ with the possible exemption of $n=133$. Our proof uses convex geometry and topology of compact Lie groups. AMS classification subject: 22E10, 52A05
Functional John and Löwner conditions for pairs of log-concave functions
本文同时推广John定理的两个方向，在泛函框架下研究对数凹函数对的最大与最小积分位置问题，统一了凸体与对数凹函数两种设定下的已知结果，并揭示John与Löwner对偶问题在函数情形下关系更复杂。
Flat grazes of convex bodies and local characterization of quadrics
本文通过局部化Blaschke定理，研究了二次曲面和椭球体的grazes平坦性特征化问题。
We define a one parameter family of positions of a convex body which interpolates between the John position and the Loewner position: for $r>0$, we say that $K$ is in maximal intersection position of radius $r$ if $\textrm{Vol}_{n}(K\cap rB_{2}^{n})\geq \textrm{Vol}_{n}(K\cap rTB_{2}^{n})$ for all $T\in SL_{n}$. We show that under mild conditions on $K$, each such position induces a corresponding isotropic measure on the sphere, which is simply a normalized Lebesgue measure on $r^{-1}K\cap S^{n-1}$. In particular, for $r_{M}$ satisfying $r_{M}^{n}κ_{n}=\textrm{Vol}_{n}(K)$, the maximal interse…
An infinitely smooth convex body in $\mathbb R^n$ is called polynomially integrable of degree $N$ if its parallel section functions are polynomials of degree $N$. We prove that the only smooth convex bodies with this property in odd dimensions are ellipsoids, if $N\ge n-1$. This is in contrast with the case of even dimensions and the case of odd dimensions with $N<n-1$, where such bodies do not exist, as it was recently shown by Agranovsky.
John Ellipsoids of Revolution
该研究通过引入旋转椭球在凸体中找最大欧几里得球与最大体积椭球的经典问题间建立插值，推导了带仿射变换限制的凸体最大体积仿射像最优性一阶条件，并揭示了极值旋转椭球的对应性质。
The Petty Conjecture for Convex Bodies of Revolution
本文证明了Petty猜想对所有n≥3维空间中的旋转凸体成立，即商式V_n(ΠK)/V_n(K)^{n-1}的最小值由椭球体取得，并刻画了等号情形。
On Certain Extremal Banach-Mazur Distances and Ader's Characterization of Distance Ellipsoids
本文证明三维空间中平行多面体与交叉多胞体是 Banach-Mazur 距离达到 $\sqrt{n}$ 的唯一极大化对象，并基于 Ader 的距离椭球刻画给出绕开体积概念的证明，同时建立平行四边形在平面问题中的唯一性与稳定性。
It is natural to ask whether the center of mass of a convex body $K\subset \mathbb{R}^n$ lies in its John ellipsoid $B_K$, i.e., in the maximal volume ellipsoid contained in $K$. This question is relevant to the efficiency of many algorithms for convex bodies. In this paper, we obtain an unexpected negative result. There exists a convex body $K\subset \mathbb{R}^n$ such that its center of mass does not lie in the John ellipsoid $B_K$ inflated $(1-C\sqrt{\frac{\log(n)} {n}})n$ times about the center of $B_K$. Moreover, there exists a polytope $P \subset \mathbb{R}^n$ with $O(n^2)$ facets whose …
A characterization of the ellipsoid through planar grazes
本文通过平面掠射特征证明，若凸体满足特定条件，则其为椭球
Ellipsoids in pseudoconvex domains II
本文针对拟凸域中的椭球问题，围绕 Griffiths 猜想展开研究，证明该猜想虽可能成立，但无法通过函子性的纤维构造从线丛 $O_{\mathbb P E}(1)$ 的正曲率厄米特度量得到向量丛 $E$ 的对应度量。
In this paper, we extend and generalize several previous works on maximal-volume positions of convex bodies. First, we analyze the maximal positive-definite image of one convex body inside another, and the resulting decomposition of the identity. We discuss continuity and differentiability of the mapping associating a body with its positive John position. We then introduce the saddle-John position of one body inside another, proving that it shares some of the properties possessed by the position of maximal volume, and explain how this can be used to improve volume ratio estimates. We investiga…
We prove that if a convex set in Cn contains two inscribed complex ellipsoid of maximal volume then one is a translate of the other. On the other hand, the circumscribed complex elipsoid of minimal volume is unique. As application we prove the complex analoge of Brunn's characterization of ellipsods.
General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the boundary of the geometric (Minkowski) sum of $k$ ellipsoids in $n$-dimensional Euclidean space. Previously this was done through iterative algorithms in which each new ellipsoid was added to an ellipsoid approximation of the sum of the previous ellipsoids. Here we provide one shot formulas to add $k$ ellipsoids directly with no intermediate approximations required. This allows us to observe a new degree of freedom in the family of ellipsoidal bounds on the geometric sum. We demonstr…
On flat shadow boundaries from point light sources and the characterization of ellipsoids
研究点光源下阴影边界平坦的凸体必为椭球体，并建立点光源照明与超平面截面对称性的对偶框架。
Given a centrally symmetric convex body $K \subset \mathbb{R}^d$ and a positive number $λ$, we consider, among all ellipsoids $E \subset \mathbb{R}^d$ of volume $λ$, those that best approximate $K$ with respect to the symmetric difference metric, or equivalently that maximize the volume of $E\cap K$: these are the maximal intersection (MI) ellipsoids introduced by Artstein-Avidan and Katzin. The question of uniqueness of MI ellipsoids (under the obviously necessary assumption that $λ$ is between the volumes of the John and the Loewner ellipsoids of $K$) is open in general. We provide a positiv…
On Bezdek's conjecture for high-dimensional convex bodies with an aligned center of symmetry
本文在高维空间中验证了Bezdek猜想，假设通过固定点的截面具有反射轴，且互补不变子空间平行于固定超平面，证明了在正交和仿射设置中成立。
A subset of a convex body $B$ containing the origin in a Euclidean space is {\it parkable in $B$} if it can be translated inside $B$ in such a manner that the translate the origin. We provide characterizations of ellipsoids and of centrally symmetric convex bodies in Euclidean spaces of dimension $\ge 3$ based on the notion of parkability, answering several questions posed by G. Bergman. The techniques used, which are based on characterizations of Hilbert spaces among finite-dimensional Banach spaces in terms of their lattices of subspaces and algebras of endomorphisms, also apply to improve a…
We show that if $d\ge 4$ is even, then one can find two essentially different convex bodies such that the volumes of their maximal sections, central sections, and projections coincide for all directions.
John-type decompositions for affinely-optimal positions of convex bodies
本文提出了一种约翰型分解，用于确定凸体在仿射最优位置下的必要条件，并应用于Banach-Mazur距离的优化问题。
In this paper we state a one-to-one connection between the maximal ratio of the circumradius and the diameter of a body (the Jung constant) in an arbitrary Minkowski space and the maximal Minkowski asymmetry of the complete bodies within that space. This allows to generalize and unify recent results on complete bodies and to derive a necessary condition on the unit ball of the space, assuming a given body to be complete. Finally, we state several corollaries, i.e. concerning the Helly dimension or the Banach-Mazur distance.
A stability version of the Blaschke-Santaló inequality and the affine isoperimetric inequality for convex bodies of dimension n>2 is proved. The first step is the reduction to the case when the convex body is o-symmetric and has axial rotational symmetry. This step works for related inequalities compatible with Steiner symmetrization. Secondly, for these convex bodies, a stability version of the characterization of ellipsoids by the fact that each hyperplane section is centrally symmetric is established.
We discuss how well a given convex body B in a real d-dimensional vector space V can be approximated by a set X for which the membership question: ``given an x in V, does x belong to X?'' can be answered efficiently (in time polynomial in d). We discuss approximations of a convex body by an ellipsoid, by an algebraic hypersurface, by a projection of a polytope with a controlled number of facets, and by a section of the cone of positive semidefinite quadratic forms. We illustrate some of the results on the Traveling Salesman Polytope, an example of a complicated convex body studied in combinato…
One of the fundamental results in Convex Geometry is Busemann's theorem, which states that the intersection body of a symmetric convex body is convex. Thus, it is only natural to ask if there is a quantitative version of Busemann's theorem, i.e., if the intersection body operation actually improves convexity. In this paper we concentrate on the symmetric bodies of revolution to provide several results on the (strict) improvement of convexity under the intersection body operation. It is shown that the intersection body of a symmetric convex body of revolution has the same asymptotic behavior ne…
The isodiametric inequality states that the Euclidean ball maximizes the volume among all convex bodies of a given diameter. We are motivated by a conjecture of Makai Jr.~on the reverse question: Every convex body has a linear image whose isodiametric quotient is at least as large as that of a regular simplex. We relate this reverse isodiametric problem to minimal volume enclosing ellipsoids and to the Dvoretzky-Rogers-type problem of finding large volume simplices in any decomposition of the identity matrix. As a result, we solve the reverse isodiametric problem for $o$-symmetric convex bodie…
Maximal volume of convex bodies avoiding random points and lacunary dilates
该研究确定了缺项序列下凸体避开随机点的最大体积渐近公式，并证明缺项性条件不可被稀疏性替代。
Convex bodies with centrally symmetric sections
本文研究Barker-Larman猜想的一个特例，证明当凸体K是中心对称且严格凸，且球B满足一定条件时，K必为椭球。
Let $d \ge 2$, and let $K \subset {\Bbb{R}}^d$ be a convex body containing the origin $0$ in its interior. In a previous paper we have proved the following. The body $K$ is $0$-symmetric if and only if the following holds. For each $ω\in S^{d-1}$, we have that the $(d-1)$-volume of the intersection of $K$ and an arbitrary hyperplane, with normal $ω$, attains its maximum if the hyperplane contains $0$. An analogous theorem, for $1$-dimensional sections and $1$-volumes, has been proved long ago by Hammer (\cite{H}). In this paper we deal with the ($(d-2)$-dimensional) surface area, or with lower…
Smooth valuations on convex bodies and finite linear combinations of mixed volumes
本文基于 Alesker 对 McMullen 猜想的解，证明了凸体上连续平移不变齐次赋值可由有限个混合体积线性组合一致逼近，且部分参数可固定为不依赖赋值的椭球体。
On random diameters of convex bodies
本文研究凸体随机截面直径的上下界，并应用于p-椭球体，解决了信息复杂性理论中关于随机信息替代最优信息的自然猜想，给出最优二分法。
In this paper we prove the Gromov--Milman conjecture (the Dvoretzky type theorem) for homogeneous polynomials on $\mathbb R^n$, and improve bounds on the number $n(d,k)$ in the analogous conjecture for odd degrees $d$ (this case is known as the Birch theorem) and complex polynomials. We also consider a stronger conjecture on the homogeneous polynomial fields in the canonical bundle over real and complex Grassmannians. This conjecture is much stronger and false in general, but it is proved in the cases of $d=2$ (for $k$'s of certain type), odd $d$, and the complex Grassmannian (for odd and even…
On the volume of convolution bodies in the plane
本文证明在平面中固定体积的凸体上，任意δ∈(0,1)的δ-卷积体体积最大化者并非椭球，这一结果与δ→1⁻时由Petty投影不等式决定的极限行为相矛盾。
Convex bodies with algebraic section volume functions
本文研究凸体截面体积函数的代数性质，证明在光滑严格凸体中，椭球体可由特定多项式代数方程完全刻画，从而推广了多项式可积区域的相关结论。
It is shown by Makai, Martini, and Ódor that a convex body $K\subset\mathbb{R}^n$, all of whose maximal sections pass through the origin, must be origin-symmetric. We prove a stability version of this result. We also discuss a theorem of Koldobsky and Shane about determination of convex bodies by fractional derivatives of the parallel section function, and establish the corresponding stability result.
On separably integrable symmetric convex bodies
本文研究无限光滑对称凸体的 $k$-可分可积性，通过分析其 $k$ 维各向同性体积函数的变量分离性质，给出了此类凸体的完整分类：当 $d-k$ 为偶数时为椭球体，为奇数时为欧几里得球，推广了多项式可积凸体的分类。
Convex bodies with equipotential circles
本文研究了具有等势圆的凸体，证明其对称中心与等势圆中心一致，并在特定条件下证明该凸体为圆，同时探讨了三维空间中的椭球和球的特性。
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.
On characteristic properties of the ellipsoid in terms of circumscribed cones of a convex body
本文通过研究凸体的外接锥特征，证明了在特定条件下，凸体或其子集为椭球的条件。
This article derives closed-form parametric formulas for the Minkowski sums of convex bodies in d-dimensional Euclidean space with boundaries that are smooth and have all positive sectional curvatures at every point. Under these conditions, there is a unique relationship between the position of each boundary point and the surface normal. The main results are presented as two theorems. The first theorem directly parameterizes the Minkowski sums using the unit normal vector at each surface point. Although simple to express mathematically, such a parameterization is not always practical to obtain…
Gaussian Zonoids, Gaussian determinants and Gaussian random fields
本文研究非中心高斯向量对应的 Vitale zonoid，用显式椭球逼近高斯 zonoid，并据此估计非中心高斯矩阵行列式期望及非中心高斯随机场零集的小方差渐近。
Notes on non-separable arrangements of convex bodies
本文研究了弱非分离凸多面体家族的性质，探讨了其稳定性及最大家族的结构，并扩展了弱k-不可穿透家族的概念。
In 1969, Vic Klee asked whether a convex body is uniquely determined (up to translation and reflection in the origin) by its inner section function, the function giving for each direction the maximal area of sections of the body by hyperplanes orthogonal to that direction. We answer this question in the negative by constructing two infinitely smooth convex bodies of revolution about the $x_n$-axis in $\R^n$, $n\ge 3$, one origin symmetric and the other not centrally symmetric, with the same inner section function. Moreover, the pair of bodies can be arbitrarily close to the unit ball.
In this paper we determine new upper bounds for the maximal density of translative packings of superballs in three dimensions (unit balls for the $l^p_3$-norm) and of Platonic and Archimedean solids having tetrahedral symmetry. Thereby, we improve Zong's recent upper bound for the maximal density of translative packings of regular tetrahedra from $0.3840\ldots$ to $0.3745\ldots$, getting closer to the best known lower bound of $0.3673\ldots$ We apply the linear programming bound of Cohn and Elkies which originally was designed for the classical problem of densest packings of round spheres. The…
Upper Bounds on Covering Minima of Convex Bodies
本文给出了凸体覆盖极值的两个新上界，基于投影和线性子空间的覆盖极值。结果适用于两个凸体的直接和，推广了之前关于直接和的覆盖半径和格宽的研究，并应用于标准终端单纯形，缩小了Gonzaléz Merino和Schymura(2017)猜想中的上下界差距。
We provide general inequalities that compare the surface area S(K) of a convex body K in ${\mathbb R}^n$ to the minimal, average or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of K. We examine separately the dependence of the constants on the dimension in the case where K is in some of the classical positions or K is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.
Convex bodies with sections with hyperplanes of symmetry
本文研究凸体截面具有平面对称性的条件，证明若某点为 Larman 点而非旋转点且所有截面均具有平行于固定超平面的对称平面，则该凸体必为旋转椭球。
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