B. Klartag
Given an arbitrary convex symmetric n-dimensional body, we construct a natural and non-trivial continuous map which associates ellipsoids to ellipsoids, such that the Lowner-John ellipsoid of the body is its unique fixed point. A new characterization of the Lowner-John ellipsoid is obtained, and we also gain information regarding the contact points of inscribed ellipsoids with the body.
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The aim of this paper is to develop the $L_p$ John ellipsoid for the geometry of log-concave functions. Using the results of the $L_p$ Minkowski theory for log-concave function established in \cite{fan-xin-ye-geo2020}, we characterize the $L_p$ John ellipsoid for log-concave function, and establish some inequalities of the $L_p$ John ellipsoid for log-concave function. Finally, the analog of Ball's volume ratio inequality for the $L_p$ John ellipsoid of log-concave function is established.
Using an idea of Voronoi in the geometric theory of positive definite quadratic forms, we give a transparent proof of John's characterization of the unique ellipsoid of maximum volume contained in a convex body. The same idea applies to the 'hard part' of a generalization of John's theorem and shows the difficulties of the corresponding 'easy part'.
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 …
We introduce a new way of representing logarithmically concave functions on $\mathbb{R}^{d}$. It allows us to extend the notion of the largest volume ellipsoid contained in a convex body to the setting of logarithmically concave functions as follows. For every $s>0$, we define a class of non-negative functions on $\mathbb{R}^{d}$ derived from ellipsoids in $\mathbb{R}^{d+1}$. For any log-concave function $f$ on $\mathbb{R}^{d}$, and any fixed $s>0$, we consider functions belonging to this class, and find the one with the largest integral under the condition that it is pointwise less than or eq…
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…
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 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.
Dual $L_p$ John ellipsoids for general measures
本文在对偶加权$L_p$ Brunn-Minkowski理论框架下，将对偶$L_p$ John椭球推广到含高斯测度的一般测度情形，证明了对应优化问题解的相关性质并给出相关几何结果。
Focal Sets of the John Ellipsoid of a Simplex from Its Vertices
本文从单纯形顶点构造三次向量场，证明其雅可比矩阵在 John 椭球焦点集上具有重特征值，并给出各维例子，平面情形与 Siebeck–Marden 定理相关。
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。
From the Steiner Inellipse to the John Ellipsoid of a Simplex: A Corner-Volume Characterization
本文将三角形中Steiner内切椭圆的角面积刻画推广到任意维单纯形，证明点位于John椭球边界当且仅当各角单纯形体积的\\(2/n\\)次方之和等于\\(V^{2/n}/n\\)，并联系中心二阶矩。
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…
On the Löwner-John Ellipsoids of the Metric Polytope
本文研究度量多面体的椭球逼近问题，给出了其内部最大体积椭球的显式描述，证明内外Löwner-John椭球的膨胀/收缩因子为Θ(n)，优于通用定理给出的O(n²)上界。
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。
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。
Recall that a convex body $K$ is in John's position if the unit Euclidean ball is the maximal volume ellipsoid contained in $K$. Approximating convex body in John's position by polytopes we obtain the following results. 1. Let $n>R_n\ge 1$ be a sequence such that $\lim_{n\rightarrow \infty} \frac{R_n}{n}=0$. For a sufficiently large $n$, we can construct a convex body $K\subset \mathbb{R}^n$ in John's position such that there is no $P$, polytope with a polynomial number of facets in $n$ such that $K\subset P\subset R_nK$; 2. For a sufficiently large $n$, there is a convex body $K\subset \mathb…
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。
As a discrete counterpart to the classical John theorem on the approximation of (symmetric) $n$-dimensional convex bodies $K$ by ellipsoids, Tao and Vu introduced so called generalized arithmetic progressions $P(A,b)\subset Z^n$ in order to cover (many of) the lattice points inside a convex body by a simple geometric structure. Among others, they proved that there exists a generalized arithmetic progressions $P(A,b)$ such that $P(A,b)\subset K\cap Z^n\subset P(A,O(n)^{3n/2}b)$. Here we show that this bound can be lowered to $n^{O(\ln n)}$ and study some general properties of so called unimodul…
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。
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。
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 Sections of Convex Bodies in John's Position and of Generalised $B_p^n$ Balls
本文通过结合几何Brascamp-Lieb不等式与广义Parseval型恒等式，改进了K.Ball的论证，为约翰位置下凸体截面体积提供了精确估计，并推广了Meyer-Pajor和Koldobsky关于$B_p^n$球体极值截面的经典估计。
Improved bounds for a discrete John-type theorem
研究离散约翰型定理覆盖问题，通过证明在维度\(n\)中凸进展可被无限恰当广义算术进展覆盖，且其大小在原集合基数\(O(n)^{2n}\)倍内，改进了之前的\(O(n)^{3n}\)倍，同时指出\(\Omega(n)^n\)阶损失不可避免。
We prove a conjecture of B. Grünbaum stating that the set of affine invariant points of a convex body equals to the set of points invariant under all affine linear symmetries of the convex body. As a consequence we give a short proof on the fact that the affine space of affine linear points is infinite dimensional. In particular, we show that the set of affine invariant points with no dual is of second category. We investigate extremal cases for a class of symmetry measures. We show that the center of the John and Löwner ellipsoid can be far apart and we give the optimal order for the extremal…
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…
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