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arXiv · 2410.14962

Asymptotic theory of $C$-pseudo-cones

Xudong Wang·Wenxue Xu·Jiazu Zhou·Baocheng Zhu
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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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Xudong Wang, Wenxue Xu, Jiazu Zhou, Baocheng Zhu. 2024-10-19. Asymptotic theory of $C$-pseudo-cones. https://arxiv.org/abs/2410.14962

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