On the metric projection onto a convex set: reverse Hölder inequalities and upper bounds

Abstract

We study the $L^p(\hat{I}¼)$-norm of the metric projection onto a closed, convex set $C\subset {\mathbf{R}}^n$ when $\hat{I}¼$ is the uniform measure on the sphere or the standard Gaussian measure on ${\mathbf{R}}^n$. Up to universal constants, we determine the optimal reverse Hölder inequalities (i.e., $L^q-L^p$ estimates for $q>p$) for both settings and for all $1\le p<q\le \infty$. The optimal constants in these inequalities depend polynomially on the dimension $n$. We establish upper bounds for the expected norm of the metric projection for a wide class of probability measures. Our inequalities improve and extend previous results of S. Chatterjee.

37 pages; comments welcome!