The dual Orlicz curvature measures for log-concave functions and their related Minkowski problems

TL;DR

This paper introduces dual Orlicz curvature measures for log-concave functions, deriving variational formulas and establishing partial existence results for the associated Minkowski problem, extending convex geometric concepts to a functional setting.

Contribution

It develops a new family of dual Orlicz curvature measures for log-concave functions and proves partial existence results for the related Minkowski problem, bridging convex geometry and functional analysis.

Findings

Derived variational formulas for Orlicz moments under Asplund sum.

Introduced dual Orlicz curvature measures for log-concave functions.

Established partial existence results for the functional dual Orlicz Minkowski problem.

Abstract

The variation of a class of Orlicz moments with respect to the Asplund sum within the class of log-concave functions is demonstrated. Such a variational formula naturally leads to a family of dual Orlicz curvature measures for log-concave functions. They are functional analogs of dual (Orlicz) curvature measures for convex bodies. Partial existence results for the functional dual Orlicz Minkowski problem are shown.