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arXiv · 1811.06191

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

Johannes Hosle
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Abstract

In this manuscript, we study the inequalities between measures of convex bodies implied by comparison of their projections and sections. Recently, Giannopoulos and Koldobsky proved that if convex bodies $K, L$ satisfy $|K|\theta^{\perp}| \le |L \cap \theta^{\perp}|$ 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 $|K| \le \sqrt{n}|L|$. The condition we consider is weaker than both the conditions $|K \cap \theta^{\perp}| \le |L \cap \theta^{\perp}|$ and $|K|\theta^{\perp}| \le |L|\theta^{\perp}|$ for all $\theta \in S^{n-1}$ that appear in the Busemann-Petty and Shephard problems respectively. Secondly, we appropriately extend the result of Giannopoulos and Koldobsky to various classes of measures possessing concavity properties, including log-concave measures.

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Johannes Hosle. 2018-11-15. On the Comparison of Measures of Convex Bodies via Projections and Sections. https://arxiv.org/abs/1811.06191

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