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arXiv · 2106.13212

General Measure Extensions of Projection Bodies

Dylan Langharst·Michael Roysdon·Artem Zvavitch
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Abstract

The inequalities of Petty and Zhang are affine isoperimetric-type inequalities providing sharp bounds for $\text{vol}^{n-1}_{n}(K)\text{vol}_n(\Pi^\circ K),$ where $\Pi K$ is a projection body of a convex body $K$. In this paper, we present a number of generalizations of Zhang's inequality to the setting of arbitrary measures. In addition, we introduce extensions of the projection body operator $\Pi$ to the setting of arbitrary measures and functions, while providing associated inequalities for this operator; in particular, Zhang-type inequalities. Throughout, we apply shown results to the standard Gaussian measure.

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Dylan Langharst, Michael Roysdon, Artem Zvavitch. 2021-06-24. General Measure Extensions of Projection Bodies. https://doi.org/10.1112/plms.12477

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