The $L_{p}$ Gaussian Minkowski problem for $C$-pseudo-cones

TL;DR

This work resolves the normalized $L_p$ Gaussian Minkowski problem for $C$-pseudo-cones, extending the Gaussian surface area and Gaussian cone settings to the $L_p$ framework. It develops a variational approach based on Wulff shapes and Gaussian measures, proving existence for $p>0$ via a max–min principle and a first-variation condition that yields a representation of the prescribed measure as a multiple of the $L_p$ Gaussian surface area measure. For $p<0$, a reciprocal variational functional is introduced to recover existence, with a parallel approximation argument to extend to the full domain. Uniqueness is shown for $p o(0,1]$ under a volume constraint, leveraging the log-concavity of the Gaussian measure and an $L_p$ mixed-volume inequality, while non-uniqueness occurs for $p<n$ in the unconstrained setting via explicit constructions. Overall, the paper broadens Gaussian Minkowski theory to $C$-pseudo-cones, providing both variational tools and a complete existence/uniqueness landscape across the sign of $p$.

Abstract

The $L_{p}$ Gaussian Minkowski problem for $C$-pseudo-cones is studied in this paper, and the existence and uniqueness results are established. This extends our previous work on the Minkowski problem for $C$-pseudo-cones with respect to the Gaussian surface area measure ($p=1$) and the Gaussian cone measure ($p=0$).

Theorems & Definitions (39)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 3.1
• Definition 3.2