On Projections of Metric Spaces

Let $X$ be a metric space and let $\mu$ be a probability measure on it. Consider a Lipschitz map $T:X\to \text{Rn}$, with Lipschitz constant $\leq 1$\mathrm{.}Thenonecanaskwhethertheimage$TX$ can have large projections on many directions. For a large class of spaces $X$, we show that there are directions $\varphi \in \text{nsphere}$ 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 $X$.