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Asymptotic theory of $C$-pseudo-cones

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arxiv 2410.14962 v2 pith:BPOKBX2V submitted 2024-10-19 math.MG

Xudong Wang , Wenxue Xu , Jiazu Zhou , Baocheng Zhu This is my paper · ORCID
classification math.MG
keywords asymptoticpseudo-conesthetaweightedbrunn-minkowskiinequalityminkowskinon-degenerated
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abstract

In this paper, we study the non-degenerated $C$-pseudo-cones which can be uniquely decomposed into the sum of a $C$-asymptotic set and a $C$-starting point. Combining this with the novel work in \cite{Schneider-A_weighted_Minkowski_theorem}, we introduce the asymptotic weighted co-volume functional $T_\Theta(E)$ of the non-degenerated $C$-pseudo-cone $E$, which is also a generalized function with the singular point $o$ (the origin). Using our convolution formula for $T_\Theta(E)$, we establish a decay estimate for $T_\Theta(E)$ at infinity and present some interesting results. As applications of this asymptotic theory, we prove a weighted Brunn-Minkowski type inequality and study the solutions to the weighted Minkowski problem for pseudo-cones. Moreover, we pose an open problem regarding $T_\Theta(E)$, which we call the asymptotic Brunn-Minkowski inequality for $C$-pseudo-cones.

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Cited by 2 Pith papers

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

  1. The Gaussian Minkowski-type problems for $C$-pseudo-cones

    math.MG 2025-01 conditional novelty 7.0 of 10

    New existence and conditional uniqueness theorems for Gaussian Minkowski and log-Minkowski problems on C-pseudo-cones.

  2. The Gaussian-Minkowski problem for $C$-pseudo-cones

    math.FA 2024-12 reject novelty 6.0 of 10

    For every finite measure on the polar directions of a pointed cone, there exists a C-pseudo-cone whose Gaussian surface area measure equals that measure, with Gaussian co-volume no more than half the cone's.

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