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arXiv · 2503.00949

On general versions of the Petty projection inequality

Francisco Marín Sola
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Abstract

The classical Petty projection inequality is an affine isoperimetric inequality which constitutes a cornerstone in the affine geometry of convex bodies. By extending the polar projection body to an inter-dimensional operator, Petty's inequality was generalized to the so-called $(L_p,Q)$ setting, where $Q$ is an $m$-dimensional compact convex set. In this work, we further extend the $(L_p,Q)$ Petty projection inequality to the broader realm of rotationally invariant measures with concavity properties, namely, those with $\gamma$-concave density (for $\gamma\geq-1/nm$). Moreover, when $p=1$, and motivated by a contemporary empirical reinterpretation of Petty's result, we explore empirical analogues of this inequality.

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Francisco Marín Sola. 2025-03-02. On general versions of the Petty projection inequality. https://arxiv.org/abs/2503.00949

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