Dual curvature measures on convex functions and associated Minkowski problems

TL;DR

This paper extends dual curvature measures to convex functions, formulates a Minkowski problem for these measures, and proves existence and uniqueness results under certain conditions, advancing the understanding of convex geometric analysis.

Contribution

It introduces a new extension of dual curvature measures to convex functions and establishes existence and uniqueness results for the associated Minkowski problem.

Findings

Existence of solutions for the functional dual Minkowski problem when q ≤ 0.

Uniqueness of solutions under certain assumptions.

Extension of dual curvature measures from convex bodies to convex functions.

Abstract

In this paper, the $q$-th dual curvature measure is extended to convex functions and the associated Minkowski problem is posed. A special case includes the $q$-th dual curvature measure of convex bodies which defined by Huang, Lutwak, Yang and Zhang. Existence for the functional dual Minkowski problem is showed when $q\leq0$ and the uniqueness part is obtained with some assumptions.