# Width of l^p balls

Antoine Gournay (LM-Orsay)

Arxiv ID: 0711.3081•Last updated: 1/5/2021

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.

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