Asymptotic theory of $C$-pseudo-cones

TL;DR

This paper develops an asymptotic theory for $C$-pseudo-cones, introducing a weighted co-volume functional, establishing decay estimates, and applying these results to inequalities and problems in convex geometry.

Contribution

It introduces the asymptotic weighted co-volume functional for $C$-pseudo-cones and applies it to derive new inequalities and solve the weighted Minkowski problem.

Findings

Established decay estimates for the weighted co-volume functional.

Proved a weighted Brunn-Minkowski type inequality.

Analyzed solutions to the weighted Minkowski problem for pseudo-cones.

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.