On projections of metric spaces

Abstract

Let $X$ be a metric space and let $\mu$ be a probability measure on it. Consider a Lipschitz map $T: X \rightarrow \mathbb{R}^n$, 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 $\phi \in \mathbb S^{n-1}$ 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 $X$.