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arXiv · 1904.10427

A Convex Body Associated to the Busemann Random Simplex Inequality and the Petty conjecture

Julián Eduardo Haddad
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Abstract

Given $L$ a convex body, the $L_p$-Busemann Random Simplex Inequality is closely related to the centroid body $\Gamma_p L$ for $p=1$ and $2$, and only in these cases it can be proved using the $L_p$-Busemann-Petty centroid inequality. We define a convex body $N_p L$ and prove an isoperimetric inequality for $(N_p L)^\circ$ that is equivalent to the $L_p$-Busemann Random Simplex Inequality. As applications, we give a simple proof of a general functional version of the Busemann Random Simplex Inequality and study a dual theory related to Petty's conjectured inequality. More precisely, we prove dual versions of the $L_p$-Busemann Random Simplex Inequality for sets and functions by means of the $p$-affine surface area measure, and we prove that the Petty conjecture is equivalent to an $L_1$-Sharp Affine Sobolev-type inequality that is stronger than (and directly implies) the Sobolev-Zhang inequality.

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BibTeXRIS

Julián Eduardo Haddad. 2019-04-23. A Convex Body Associated to the Busemann Random Simplex Inequality and the Petty conjecture. https://doi.org/10.1016/j.jfa.2021.109118

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