Width of l^p balls

TL;DR

This paper investigates the Urysohn width of metric spaces, providing bounds and estimations using various mathematical tools, and offers a complete characterization in three dimensions for certain p-values.

Contribution

It introduces new bounds for Urysohn width using Hadamard matrices, Borsuk-Ulam theorem, and other methods, and characterizes the width in dimension three for 1 ≤ p ≤ 2.

Findings

Bounds on Urysohn width using Hadamard matrices and Borsuk-Ulam theorem

Complete description of width in 3D for 1 ≤ p ≤ 2

Estimates for diameters of sets not contained in a hemisphere

Abstract

We say a map f:X \to Y is an \epsilon-embedding if it is continuous and the diameter of the fibres is less than \epsilon. This type of maps is used in the notion of Urysohn width (sometimes referred to as Alexandrov width), a_n(X). It is the smallest real number such that there exists an \epsilon-embedding from X to a n-dimensional polyhedron. Surprisingly few estimations of these numbers can be found, and one of the aims of this paper is to present some. Following Gromov, we take the slightly different point of view by looking at the smallest dimension n for which there exists a \epsilon-embedding to a polyhedron of dimension n. While bounds are obtained using Hadamard matrices, the Borsuk-Ulam theorem, the filling radius of spheres, and lower bounds for the diameter of sets of n+1 points not contained in a hemisphere (obtained by methods very close to those of Ivanov and Pichugov). We are also able to give a complete description in dimension 3 for 1 \leq p \leq 2.