Weighted Minkowski's Existence Theorem and Projection Bodies

TL;DR

This paper extends Minkowski's existence theorem to a broad class of weighted measures, introduces weighted projection bodies, and explores an isomorphic Shephard problem relating measures and convex bodies.

Contribution

It generalizes Minkowski's theorem for measures with continuous density and studies weighted projection bodies and their properties.

Findings

Existence of symmetric convex bodies for a large class of measures.

Classification of weighted projection bodies.

Solution to the isomorphic Shephard problem in specific cases.

Abstract

The Brunn-Minkowski Theory has seen several generalizations over the past century. Many of the core ideas have been generalized to measures. With the goal of framing these generalizations as a weighted Brunn-Minkowski theory, we prove the Minkowski existence theorem for a large class of Borel measures with continuous density, denoted by $\Lambda^n$: for $\nu$ a finite, even Borel measure on the unit sphere and even $\mu\in\Lambda^n$, there exists a symmetric convex body $K$ such that $$d\nu(u)=c_{\mu,K}dS^{\mu}_{K}(u),$$ where $c_{\mu,K}$ is a quantity that depends on $\mu$ and $K$ and $dS^{\mu}_{K}(u)$ is the surface area-measure of $K$ with respect to $\mu$. Examples of measures in $\Lambda^n$ are homogeneous measures (with $c_{\mu,K}=1$) and probability measures with radially decreasing densities (e.g. the Gaussian measure). We will also consider weighted projection bodies $\Pi_\mu K$ by classifying them and studying the isomorphic Shephard problem: if $\mu$ and $\nu$ are even, homogeneous measures with density and $K$ and $L$ are symmetric convex bodies such that $\Pi_{\mu} K \subset \Pi_{\nu} L$, then can one find an optimal quantity $\mathcal{A}>0$ such that $\mu(K)\leq \mathcal{A}\nu(L)$? Among other things, we show that, in the case where $\mu=\nu$ and $L$ is a projection body, $\mathcal{A}=1$.