On Projections of Metric Spaces

Original · EN

Let X be a metric space and let μ be a probability measure on it. Consider a Lipschitz map T: X →, with Lipschitz constant ≤ 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 ϕ∈ 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.