On the metric projection onto a convex set: reverse H\"older inequalities and upper bounds

Abstract

We study the $L^p(\mu)$-norm of the metric projection onto a closed, convex set $C \subset \mathbf{R}^n$ when $\mu$ 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 H\"older inequalities (i.e., $L^q-L^p$ estimates for $q>p$) for both settings and for all $1 \leq p<q \leq \infty$. 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.