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The $L_p$-Minkowski problem with super-critical exponents

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arxiv 2203.05099 v1 pith:B6D7ZGM2 submitted 2022-03-10 math.AP

Qiang Guang , Qi-Rui Li , Xu-Jia Wang This is my paper · ORCID
classification math.AP
keywords problemminkowskisuper-criticalcaseexistenceexponentstopologicalapplications
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abstract

The $L_p$-Minkowski problem deals with the existence of closed convex hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed $p$-area measures. It extends the classical Minkowski problem and embraces several important geometric and physical applications. The Existence of solutions has been obtained in the sub-critical case $p>-n-1$, but the problem remains widely open in the super-critical case $p<-n-1$. In this paper, we introduce new ideas to solve the problem for all the super-critical exponents. A crucial ingredient in our proof is a topological method based on the calculation of the homology of a topological space of ellipsoids.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Compactness of the $L_p$ dual Minkowski problem in $\mathbb{R}^3$

    math.AP 2025-05 conditional novelty 7.0 of 10

    For convex bodies in R^3, bounded L_p qth dual curvature with p in [0,1) and q>2+p forces a uniform diameter upper bound and volume lower bound.

  2. The horospherical $p$-Christoffel-Minkowski and prescribed $p$-shifted Weingarten curvature problems in hyperbolic space

    math.DG 2024-11 conditional novelty 7.0 of 10

    Smooth even strictly horospherically convex solutions exist for the horospherical p-Christoffel-Minkowski problem and the new p-shifted Weingarten problem in hyperbolic space for p≥−n, under convexity bounds on f.

  3. 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.

  4. Minkowski Problems for Geometric Measures

    math.MG 2025-02 unverdicted novelty 2.0 of 10

    A comprehensive survey that organizes the Minkowski problems of convex geometry into a unified framework based on geometric measures as differentials of global geometric invariants.

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