On Projections of Metric Spaces

TL;DR

This paper investigates how Lipschitz maps from metric spaces to Euclidean spaces project onto various directions, revealing that for many spaces, significant projections are limited on average, with bounds related to dimension and spectral properties.

Contribution

It establishes bounds on the average projection size of Lipschitz images of metric spaces, linking geometric projection behavior to spectral characteristics of the space.

Findings

Existence of directions with small average projection for Lipschitz images

Bounds depend on the dimension and Laplacian eigenvalues of the space

Results apply to a large class of metric spaces

Abstract

Let $X$ be a metric space and let $\mu$ be a probability measure on it. Consider a Lipschitz map $T: X \rightarrow \Rn$, with Lipschitz constant $\leq 1$. Then one can ask whether the image $TX$ can have large projections on many directions. For a large class of spaces $X$, we show that there are directions $\phi \in \nsphere$ on which the projection of the image $TX$ is small on the average, with bounds depending on the dimension $n$ and the eigenvalues of the Laplacian on $X$.