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

TL;DR

This work extends Gaussian Minkowski-type theory to unbounded convex sets called $C$-pseudo-cones by introducing the Gaussian surface area measure $S_{ ext{γ}}$ and the Gaussian cone measure $C_{ ext{γ}}$ on $oldsymbol{ abla}=S^{n-1}igcap ext{int} C^{ullet}$. It establishes existence for the Gaussian Minkowski and Gaussian log-Minkowski problems in this setting, up to a normalization constant due to non-homogeneity, and shows non-uniqueness in general while proving uniqueness under a fixed Gaussian volume $oldsymbol{ abla}(K)$. The paper develops variational frameworks based on Gaussian volume and logarithmic families of Wulff shapes to derive core identities and to handle the non-homogeneous nature of Gaussian space, enabling both problems to be solved for broad classes of measures. Together, these results extend Brunn–Minkowski-type theory to $C$-pseudo-cones in Gaussian space and clarify how normalization and non-homogeneity influence existence and uniqueness.

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.

Theorems & Definitions (58)

• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• proof
• Lemma 3.2