The width-volume inequality

TL;DR

This paper establishes a width-volume inequality in Euclidean space, providing bounds on the k-width of sets and implications for the dilation of degree 1 maps between domains, with applications to rectangles.

Contribution

It introduces a new width-volume inequality in Euclidean space and applies it to estimate minimal dilation of degree 1 maps between rectangles, improving understanding of geometric mappings.

Findings

Proved a width-volume inequality with bounds depending on volume and dimension.

Provided lower bounds for the k-dilation of degree 1 maps between domains.

Showed examples where linear maps are significantly less optimal than the minimal possible dilation.

Abstract

We prove that a bounded open set U in Euclidean n-space has k-width less than C(n) Volume(U)^{k/n}. Using this estimate, we give lower bounds for the k-dilation of degree 1 maps between certain domains in Euclidean space. In particular, we estimate the smallest (n-1)-dilation of any degree 1 map between two n-dimensional rectangles. For any pair of rectangles, our estimate is accurate up to a dimensional constant C(n). We give examples in which the (n-1)-dilation of the linear map is bigger than the optimal value by an arbitrarily large factor.