Weighted Minkowski's Existence Theorem and Projection Bodies

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, $\mathcal{A}=1$.