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Asymptotic theory of $C$-pseudo-cones

Xudong Wang, Wenxue Xu, Jiazu Zhou, Baocheng Zhu

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 $T_Θ(E)$ 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 $T_Θ(E)$, we establish a decay estimate for $T_Θ(E)$ 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 $T_Θ(E)$, which we call the asymptotic Brunn-Minkowski inequality for $C$-pseudo-cones.

Asymptotic theory of $C$-pseudo-cones

Abstract

In this paper, we study the non-degenerated -pseudo-cones which can be uniquely decomposed into the sum of a -asymptotic set and a -starting point. Combining this with the novel work in \cite{Schneider-A_weighted_Minkowski_theorem}, we introduce the asymptotic weighted co-volume functional of the non-degenerated -pseudo-cone , which is also a generalized function with the singular point (the origin). Using our convolution formula for , we establish a decay estimate for 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 , which we call the asymptotic Brunn-Minkowski inequality for -pseudo-cones.

Paper Structure

This paper contains 7 sections, 32 theorems, 225 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background: Push-forward measure and the Co-area formula
  3. Asymptotic analysis of $C$-pseudo-cones
  4. Finiteness of the asymptotic weighted co-volume
  5. Integral formulas and convolution formulas
  6. The Brunn-Minkowski type inequality for $C$-pseudo-cones
  7. The weighted Minkowski problem for $q\in[0,n-1)$

Key Result

Theorem 1

Let $C$ be a pointed closed convex cone in $\mathbb{R}^n$ with non-empty interior and $o\notin E\subset\mathbb{R}^n$ be a $C$-pseudo-cone, then the following two statements are equivalent: $(i)$ There is a point $z\in C$ such that $h_E(v)=\langle z,v\rangle$ for all $v\in\partial\Omega_{C^\circ}$; $

Figures (1)

  • Figure 1: The degenerated $C$-pseudo-cone in $\mathbb{R}^3$

Theorems & Definitions (70)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1: Weighted theory v.s. dual theory
  • Theorem 4
  • Definition 1: see Artstein-Avidan-A_zoo_of_dualitiesSchneider-Pseudo_conesSchneider-A_weighted_Minkowski_theoremXu-Li-Leng-Dualities
  • Definition 2: see Li-Ye-Zhu-The_dual_Minkowski_problem
  • Lemma 1: see Li-Ye-Zhu-The_dual_Minkowski_problem
  • Lemma 2
  • proof
  • ...and 60 more