Uniqueness of solutions to the isotropic $L_{p}$ Gaussian Minkowski problem

TL;DR

The work addresses the uniqueness of solutions to the isotropic $L_p$ Gaussian Minkowski problem in $\mathbb{R}^{n+1}$ for the regime $-(n+1)<p<-1$, without requiring the origin-centered condition on convex bodies. It develops a spectral/variational approach based on a local Brunn–Minkowski inequality, employing a test-function framework with $X = D h$ and analyzing the Monge–Ampère constraint $h^{1-p} \frac{1}{\kappa} e^{-rac{|D h|^2}{2}} = c$ on $\mathbb{S}^n$. The main result shows that, under $R(K) \le 1$, the boundary must be a sphere, with a detailed characterization of constant-solution possibilities via a monotone function $g(t)=t^{n+1-p} e^{-t^2/2}$ and discussion of the method's limitations for $R(K)>1$ and potential extensions. The findings support the well-posedness of degree-theoretic approaches to the (non-normalized) Gaussian Minkowski problem in the non-symmetric setting and illuminate the role of Gaussian volume constraints in uniqueness.

Abstract

The uniqueness of solutions to the isotropic $L_{p}$ Gaussian Minkowski problem in $\mathbb{R}^{n+1}$ is established when $-(n+1)<p<-1$ with $n\geq 1$, without requiring the origin-centred assumption on convex bodies.

Theorems & Definitions (12)

• Theorem 1.1
• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• proof : Proof of Theorem \ref{['MTH']}.
• Lemma A.1
• Lemma A.2
• proof