A Characterization of Functional Affine Surface Areas

2025/12/09 by Baêta, Fernanda M.
#FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2512.08375

Abstract

A characterization of valuations on the space of convex Lipschitz functions whose domain is a polytope in ℝⁿ is obtained. It is shown that every upper semicontinuous, equi-affine and dually epi-translation invariant valuation can be written as a linear combination of a constant term, the volume of the domain, and a functional affine surface area. In addition, dual statements for finite-valued convex functions are established.

Citations

• Upper semicontinuous valuations on convex functions of one variable
• On the Semicontinuity of Functionals on Function Spaces
• Additive Kinematic Formulas for Functional Minkowski Vectors
• A Klain-Schneider Theorem for Vector-Valued Valuations on Convex Functions
• Weighted floating functions and weighted functional affine surface areas
• From valuations on convex bodies to convex functions
• Continuous valuations on the space of Lipschitz functions on the sphere
• Floating functions
• Functional versions of L_p-affine surface area and entropy inequalities
• The Centro-Affine Hadwiger Theorem
• Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality

Related

• Extremal affine surface areas in a functional setting
• Upper semicontinuous valuations on convex functions of one variable
• Zonal valuations on convex bodies
• Smooth valuations on convex functions
• On the p-affine surface area