The Gaussian Minkowski-type problems for $C$-pseudo-cones

TL;DR

This paper introduces Gaussian surface area and cone measures for $C$-pseudo-cones, proposing related Minkowski problems and establishing solutions' existence and uniqueness.

Contribution

It extends Gaussian Minkowski problems to $C$-pseudo-cones, providing foundational existence and uniqueness results for these new geometric measures.

Findings

Defined Gaussian surface area and cone measures for $C$-pseudo-cones

Formulated Gaussian Minkowski and log-Minkowski problems for $C$-pseudo-cones

Proved existence and uniqueness of solutions to these problems

Abstract

The Gaussian surface area measure and the Gaussian cone measure for $C$-pseudo-cones are introduced and their corresponding Gaussian Minkowski problem and Gaussian log-Minkowski problem are proposed, respectively. The existence and uniqueness of solutions to these problems for $C$-pseudo-cones are established.