Department of
Applied
Mathematics

Lipschitz mappings of discrete sets in Euclidean spaces (Jirka Matoušek prize talk)

Vojtěch Kaluža

Abstract

(joint work with Michael Dymond and Eva Kopecká)

Consider a set of n^2 points in the plane with integer coordinates. Map the set bijectively onto a square of n x n points. If one additionally asks the mapping to be Lipschitz, what is the best Lipschitz constant that one can achieve? Can the Lipschitz constant be independent of n? This question was asked by Uriel Feige in 2002 and we answer it negatively.

My aim in this talk is to explain the motivation for the question and to provide a high-level overview of our solution skipping many somewhat technical details. Although the question itself is discrete, our solution takes place largely in the continuous world. It relies on a result on a bilipschitz decomposition of certain Lipschitz mappings and on new results on the prescribed volume form equation for Lipschitz mappings; these will be explained. I will also present several related open questions. No specific prior knowledge will be assumed.