相关论文: John-type decompositions for affinely-optimal posi…
A classical consequence of the John Ellipsoid Theorem is the upper bound $\sqrt{n}$ on the Banach-Mazur distance between the Euclidean ball and any symmetric convex body in $\mathbb{R}^n$. Equality is attained for the parallelotope and the…
We consider two well-known problems: upper bounding the volume of lower dimensional ellipsoids contained in convex bodies given their John ellipsoid, and lower bounding the volume of ellipsoids containing projections of convex bodies given…
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 $n>R_n\ge 1$…
Finding a largest Euclidean ball in a given convex body $K \subset \mathbb{R}^d$ and finding a largest volume ellipsoid in $K$ are two problems of fundamentally different nature. The first is a purely Euclidean problem, where we consider…
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…
We study the way in which the Euclidean subspaces of a Banach space fit together, somewhat in the spirit of the Ka\v{s}in decomposition. The main tool that we introduce is an estimate regarding the convex hull of a convex body in John's…
We extend the invertibility principle of J. Bourgain and L. Tzafriri to operators acting on arbitrary decompositions id = \sum x_j \otimes x_j, rather than on the coordinate one. The John's decomposition brings this result to the local…
The John equation serves as the mathematical foundation of the X-ray transform, describing the intrinsic compatibility conditions that projection data must satisfy. In this paper, within three-dimensional (3D) Euclidean space, an innovative…
The famous theorem of Fritz John states that any convex body has a unique maximal volume inscribed ellipsoid, known as the John Ellipsoid. Computing the John Ellipsoid is a fundamental problem in convex optimization. In this paper, we focus…
John's inclusion states that a convex body in $\mathbb{R}^d$ can be covered by the $d$-dilation of its maximal volume ellipsoid. We obtain a certain John-type inclusion for log-concave functions. As a byproduct of our approach, we establish…
Many years ago John Tyrell a lecturer at King's college London challenged his Ph.D. students with the following puzzle: show that there is a unique triangle of minimal perimeter with exactly one vertex to lie on one of three given lines,…
Coverings of convex bodies have emerged as a central component in the design of efficient solutions to approximation problems involving convex bodies. Intuitively, given a convex body $K$ and $\epsilon> 0$, a covering is a collection of…
In this paper we are interested in "optimal" universal geometric inequalities involving the area, diameter and inradius of convex bodies. The term "optimal" is to be understood in the following sense: we tackle the issue of…
Approximating convex bodies is a fundamental problem in geometry. Given a convex body $K$ in $\mathbb{R}^d$ for a fixed dimension $d$, the objective is to minimize the number of facets of an approximating polytope for a given Hausdorff…
We determine certain Banach-Mazur distances involving $\ell_p$-direct sums of finite-dimensional real normed spaces and related cone constructions of convex bodies. Using a recent characterization of the optimal Banach-Mazur position with…
John's fundamental theorem characterizing the largest volume ellipsoid contained in a convex body $K$ in $\mathbb{R}^d$ has seen several generalizations and extensions. One direction, initiated by V. Milman is to replace ellipsoids by…
We characterize the solution of a broad class of convex optimization problems that address the reconstruction of a function from a finite number of linear measurements. The underlying hypothesis is that the solution is decomposable as a…
The paper suggests a new --- to the best of the author's knowledge --- characterization of decisions which are optimal in the multi-objective optimization problem with respect to a definite proper preference cone, a Euclidean cone with a…
We show that the non-symmetric Banach-Mazur distance between two convex bodies $K_1, K_2 \subseteq \mathbb{R}^n$ satisfies $$ d_{BM}(K_1, K_2) \leq C n \cdot \log^{\alpha} (n+1), $$ for universal constants $C, \alpha > 0$. This improves…
We revisit an ingenious argument of K. Ball to provide sharp estimates for the volume of sections of a convex body in John's position. Our technique combines the geometric Brascamp-Lieb inequality with a generalised Parseval-type identity.…