相关论文: Intersection of unit balls in classical matrix ens…
Volume estimates of metric balls in manifolds find diverse applications in information and coding theory. In this paper, some new results for the volume of a metric ball in unitary group are derived via various tools from random matrix…
The purpose of this note is to study the asymptotic volume of intersections of unit balls associated with two norms in $\mathbb{R}^n$ as their dimension $n$ tends to infinity. A general framework is provided and then specialized to the…
In this survey, we discuss volumetric and combinatorial results concerning (mostly finite) intersections or unions of balls (mostly of equal radii) in the $d$-dimensional real vector space, mostly equipped with the Euclidean norm. Our first…
The volume of the unit ball of the Lebesgue sequence space $\ell_p^m$ is very well known since the times of Dirichlet. We calculate the volume of the unit ball in the mixed norm $\ell^n_q(\ell_p^m)$, whose special cases are nowadays popular…
We develop a probabilistic approach to study the volumetric and geometric properties of unit balls $\mathbb B_{q,1}^n$ of finite-dimensional Lorentz sequences spaces $\ell_{q,1}^n$. More precisely, we show that the empirical distribution of…
We give some natural sufficient conditions for balls in a metric space to have small intersection. Roughly speaking, this happens when the metric space is (i) expanding and (ii) well-spread, and (iii) a certain random variable on the…
We study the precise asymptotic volume of balls in Orlicz spaces and show that the volume of the intersection of two Orlicz balls undergoes a phase transition when the dimension of the ambient space tends to infinity. This generalizes a…
Considering a finite intersection of balls and a finite union of other balls in an Euclidean space, we propose an exact method to test whether the intersection is covered by the union. We reformulate this problem into quadratic programming…
We study a family of polytopes and their duals, that appear in various optimization problems as the unit balls for certain norms. These two families interpolate between the hypercube, the unit ball for the $\infty$-norm, and its dual…
We study the volume of unit balls $B^n_{p,q}$ of finite-dimensional Lorentz sequence spaces $\ell_{p,q}^n.$ We give an iterative formula for ${\rm vol}(B^n_{p,q})$ for the weak Lebesgue spaces with $q=\infty$ and explicit formulas for $q=1$…
We establish Central Limit Theorems for the volumes of intersections of $B_{p}^n$ (the unit ball of $\ell_p^n$) with uniform random subspaces of codimension $d$ for fixed $d$ and $n\to \infty$. As a corollary we obtain higher order…
We characterise the class of those Banach spaces in which every convex combination of slices of the unit ball intersects the unit sphere as the class of those spaces in which every convex combination of slices of the unit ball contains two…
Volume of metric balls relates to rate-distortion theory and packing bounds on codes. In this paper, the volume of balls in complex Grassmann manifolds is evaluated for an arbitrary radius. The ball is defined as a set of hyperplanes of a…
A ball polyhedron is a finite intersection of congruent balls in $\mathbb{R}^3$. These shapes arise in various contexts in discrete and convex geometry. We focus on Reuleaux polyhedra, the subclass of ball polyhedra whose centers and…
The aim of this work is to expose some asymptotic series associated to some expressions involving the volume of the n-dimensional unit ball. All proofs and the methods used for improving the classical inequalities announced in the final…
We obtain a new upper estimate on the Euclidean diameter of the intersection of the kernel of a random matrix with iid rows with a given convex body. The proof is based on a small-ball argument rather than on concentration and thus the…
A very fundamental geometric problem on finite systems of spheres was independently phrased by Kneser (1955) and Poulsen (1954). According to their well-known conjecture if a finite set of balls in Euclidean space is repositioned so that…
Let ${\mathbb E}^d$ denote the $d$-dimensional Euclidean space. The $r$-ball body generated by a given set in ${\mathbb E}^d$ is the intersection of balls of radius $r$ centered at the points of the given set. In this paper we prove the…
We study the smallest intersecting and enclosing ball problems in Euclidean spaces for input objects that are compact and convex. They link and unify many problems in computational geometry and machine learning. We show that both problems…
The Kneser-Poulsen conjecture says that if a finite collection of balls in a d-dimensional Euclidean space is rearranged so that the distance between each pair of centers does not get smaller, then the volume of the union of these balls…