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A necessary condition for the logarithmic Minkowksi problem in higher dimension

Mijia Lai, Zixiao Wang

TL;DR

This work extends a dimension-2 necessary condition for the logarithmic Minkowski problem to higher dimensions by proving that, for a convex body $K$ with centroid at the origin and cone-volume measure $\,\mu$, the inequality $n\frac{\mu(u)+\mu(-u)}{V}+(n+1)^{n-1}\left|\frac{\mu(u)-\mu(-u)}{V}\right|^n\leq 1$ holds for all directions $u\in\mathbb{S}^{n-1}$, where $V=\mu(\mathbb{S}^{n-1})$; equality characterizes oblique prisms or cones. The proof proceeds by three steps: reducing to Schwarz symmetrization, establishing the inequality for truncated cones, and reducing to truncated cones via a volume-preserving deformation that preserves key centroid/fractional-cone quantities. The result refines the subspace concentration condition and solidifies a higher-dimensional analogue of the 2D condition, with equality cases aligning with simple geometric primitives (prisms or cones) and implications for the discrete and continuous logarithmic Minkowski problems. The appendix provides detailed derivative-based verifications of the auxiliary lemmas used in the cone-case analysis.

Abstract

In this paper, we establish a necessary condition for the logarithmic Minkowski problem in higher dimensions. This result generalizes a necessary condition proposed by Liu, Lu, Sun, and Xiong in their investigation of the two-dimensional case, and also refines the so-called subspace concentration condition.

A necessary condition for the logarithmic Minkowksi problem in higher dimension

TL;DR

This work extends a dimension-2 necessary condition for the logarithmic Minkowski problem to higher dimensions by proving that, for a convex body with centroid at the origin and cone-volume measure , the inequality holds for all directions , where ; equality characterizes oblique prisms or cones. The proof proceeds by three steps: reducing to Schwarz symmetrization, establishing the inequality for truncated cones, and reducing to truncated cones via a volume-preserving deformation that preserves key centroid/fractional-cone quantities. The result refines the subspace concentration condition and solidifies a higher-dimensional analogue of the 2D condition, with equality cases aligning with simple geometric primitives (prisms or cones) and implications for the discrete and continuous logarithmic Minkowski problems. The appendix provides detailed derivative-based verifications of the auxiliary lemmas used in the cone-case analysis.

Abstract

In this paper, we establish a necessary condition for the logarithmic Minkowski problem in higher dimensions. This result generalizes a necessary condition proposed by Liu, Lu, Sun, and Xiong in their investigation of the two-dimensional case, and also refines the so-called subspace concentration condition.
Paper Structure (7 sections, 7 theorems, 55 equations, 2 figures)

This paper contains 7 sections, 7 theorems, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}
  3. Step 1: Reduction to its Schwarz symmetrization
  4. Step 2: Verification of \ref{['necessary']} for truncated cones
  5. Step 3: Reduction to a truncated cone
  6. Completion of the proof
  7. Appendix

Key Result

Theorem 1

Let $K\subset \mathbb{R}^n,(n\geq 3)$ be a convex body with centroid at the origin, $\mu$ be the associated cone volume measure. Set $V:=\mu(\mathbb{S}^{n-1})=V(K)$, then for any $u\in \mathbb{S}^{n-1}$, there holds Moreover if the equality holds for some $u$, then $K$ is Here $P$ is an $(n-1)$-dimensional convex body lying in an $(n-1)$-dimensional hyperplane $H$ perpendicular to $u$, and $\vec

Figures (2)

  • Figure 1: Left: oblique prism, Right: cone
  • Figure 2: Left: $K_0$, Middle: $K'$, Right: $K_1$

Theorems & Definitions (11)

  • Theorem 1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • proof : Proof of Lemma \ref{['L1']}
  • Lemma 3.2
  • ...and 1 more