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On the Comparison of Measures of Convex Bodies via Projections and Sections

Banff International Research Station for Mathematical Innovation and Discovery

We discuss inequalities between measures of convex bodies implied by comparison of their projections and sections. Recently, Giannopoulos and Koldobsky proved that if $K, L$ are convex bodies in $\mathbb{R}^n$ with $|K|\theta^{\perp}| \le |L\cap \theta^{\perp}|$ for all $\theta \in S^{n-1}$, then $|K| \le |L|$. Firstly, we study the reverse question: in particular, we show that if $K, L$ are origin-symmetric convex bodies in John's position with $|K \cap \theta^{\perp}| \le |L|\theta^{\perp}|$ for all $\theta \in S^{n-1}$, then $|K| \le \sqrt{n}|L|$. We also discuss an extension of the result of Giannopoulos and Koldobsky to log-concave measures and an extension of the Loomis-Whitney inequality to positively concave and positively homogeneous measures.

Mathematics; Convex and discrete geometry; Functional analysis

video/mp4

Author affiliation: UCLA

2020-08-09

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/75399

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/