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Higher order Poincare inequalities and Minkowski-type inequalities

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arxiv 2103.10627 v1 pith:R3DNVCNF submitted 2021-03-19 math.DG math.FA

Kwok-Kun Kwong This is my paper · ORCID
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We observe some higher order Poincare-type inequalities on a closed manifold, which is inspired by Hurwitz's proof of the Wirtinger's inequality using Fourier theory. We then give some geometric implication of these inequalities by applying them on the sphere. More specifically, by applying them to the support function of a convex hypersurface in the Euclidean space, we obtain some sharp Minkowski-type inequalities, such as a stability inequality for the classical Minkowski inequality and the Alexandrov-Fenchel inequality.

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  1. Uniqueness of $S_2$-isotropic solutions to the isotropic $L_p$ Minkowski problem

    math.DG 2025-09 conditional novelty 6.0 of 10

    Under a spectral-gap assumption on the Hilbert-Brunn-Minkowski operator, every S2-isotropic solution of the isotropic Lp Minkowski problem in the supercritical range p<-n is the unit ball.

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