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On the Comparison of Measures of Convex Bodies via Projections and Sections Hosle, Johannes


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.

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