Weighted Minkowski’s existence theorem and projection bodies

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 \begin{equation*} d\nu (u)=c_{\mu ,K}dS^{\mu }_{K}(u), \end{equation*} 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, $\mathcal {A}=1$.