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 $$d\nu (u)=c_{\mu ,K}d{S}_K^{\mu }(u),$$ where $c_{\mu ,K}$ is a quantity that depends on $\mu$ and $K$ and $d{S}_K^{\mu }(u)$ 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 $\mu (K)\le \mathcal{A}\nu (L)$? Among other things, we show that, in the case where $\mu =\nu$ and $L$ is a projection body, $\mathcal{A}=1$.

Abstract updated. Formally titled "Measure Theoretic Minkowski's Existence Theorem and Projection Bodies". 42-46 pages. Keywords: Minkowski's Existence Theorem, Shephard Problem, Petty Projection Inequality, Projection Body, Ehrhard's Inequality

References in corpus (4)

• The Logarithmic Minkowski Problem
• Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems
• General Measure Extensions of Projection Bodies
• Weighted Brunn-Minkowski Theory I: On Weighted Surface Area Measures

Cited by in corpus (1)

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