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The $L_p$ Minkowski problems on affine dual quermassintegrals

Youjiang Lin, Yuchi Wu

TL;DR

The paper extends the theory of affine dual curvature measures to the $L_p$ setting on affine dual quermassintegrals (ADQ) and solves the existence part of the $L_p$ Minkowski problem for ADQ data across discrete and continuous measures. It develops an $L_p$-dependent variation formula, continuity results, and inradius-type estimates, and proves existence for discrete non-symmetric data when $p>1$, for general non-symmetric data when $p>n+m(n-2)$, and for even symmetric data under a strict subspace concentration inequality when $p\ge0$; the $p=0$ case recovers the classical affine dual problem. The approach combines a Wulff-family variation, dual Radon-transform tools, and an approximation-by-discrete-measures framework to construct polytopal solutions and pass to the limit, connecting to bi-dual intersection bodies. These results broaden the affine dual Brunn-Minkowski theory and provide existence (and, in certain regimes, uniqueness) results for the ADQ-based Minkowski problem across broad measure classes.

Abstract

In this paper, {we extend the affine dual curvature measures to the $L_p$ setting and solve the existence part of the corresponding Minkowski problem for non-symmetric discrete measures when $p>1$ and for symmetric measures when $p\geq0$.} When $p=0$, the $L_0$ Minkowski problem is the affine dual Minkowski problem, which is introduced and solved by Cai-Leng-Wu-Xi in [6].

The $L_p$ Minkowski problems on affine dual quermassintegrals

TL;DR

The paper extends the theory of affine dual curvature measures to the setting on affine dual quermassintegrals (ADQ) and solves the existence part of the Minkowski problem for ADQ data across discrete and continuous measures. It develops an -dependent variation formula, continuity results, and inradius-type estimates, and proves existence for discrete non-symmetric data when , for general non-symmetric data when , and for even symmetric data under a strict subspace concentration inequality when ; the case recovers the classical affine dual problem. The approach combines a Wulff-family variation, dual Radon-transform tools, and an approximation-by-discrete-measures framework to construct polytopal solutions and pass to the limit, connecting to bi-dual intersection bodies. These results broaden the affine dual Brunn-Minkowski theory and provide existence (and, in certain regimes, uniqueness) results for the ADQ-based Minkowski problem across broad measure classes.

Abstract

In this paper, {we extend the affine dual curvature measures to the setting and solve the existence part of the corresponding Minkowski problem for non-symmetric discrete measures when and for symmetric measures when .} When , the Minkowski problem is the affine dual Minkowski problem, which is introduced and solved by Cai-Leng-Wu-Xi in [6].

Paper Structure

This paper contains 14 sections, 26 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Convex bodies and star bodies
  4. Radial Gauss image and its reverse.
  5. Intersection body
  6. $L_p$ dual curvature measures and $L_q$ dual intrinsic volume
  7. $L_p$ curvature measures of ADQ on $\mathcal{K}^n_{o}$
  8. Definitions of $L_p$ curvature measures of ADQ
  9. The variation formula on $\widetilde{C}_{p,m}^a$
  10. The continuity of affine dual curvature measures
  11. Some properties on $L_p$ curvature measures of ADQ
  12. Solution to the $L_p$ Minkowski problem on ADQ for non-symmetric discrete measures
  13. Solution to the $L_p$ Minkowski problem on ADQ for non-symmetric general measures
  14. Solution to the $L_p$ Minkowski problem on ADQ for symmetric measures

Key Result

Theorem 1.1

Let $m=1,\dots,n-1$, $p>1$ and $p\neq mn$. Let $\mu$ be a discrete measure on $S^{n-1}$ that is not concentrated on any closed hemisphere. Then there exists a polytope $P\subset \mathbb{R}^n$ containing the origin in its interior such that $\widetilde{C}^a_{p,m}(P,\cdot)=\mu$.

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Corollary 3.4
  • ...and 30 more