Polynomial local functionals on convex functions

2025/12/17 by Knoerr, Jonas
Mathematics · #Advanced Banach Space Theory #Holomorphic and Operator Theory #Advanced Differential Equations and Dynamical Systems

paper · doi:10.48550/arxiv.2512.15402

Abstract

We show that every continuous local functional on the space of finite convex functions on ℝⁿ is a valuation. This relation is used to establish a homogeneous decomposition for the class of polynomial local functionals as well as a classification of translation or rigid motion invariant polynomial local functionals. In addition we discuss implications for the compact-open topology on the space of polynomial local functionals.

Citations

• The Vectorial Hadwiger Theorem on Convex Functions
• Polynomial valuations on convex functions and their maximal extensions
• Monge-Ampère operators and valuations
• Singular valuations and the Hadwiger theorem on convex functions
• Smooth valuations on convex functions
• Valuations on convex functions and convex sets and Monge-Ampere\n operators
• Integral geometry of translation invariant functionals, II: The case of\n general convex bodies
• Integral geometry of translation invariant functionals, I: The polytopal case
• Integral representation for a class of C¹-convex functionals

Related

• Polynomial valuations on convex functions and their maximal extensions
• Valuations on convex functions and convex sets and Monge-Ampere\n operators
• Smooth valuations on convex functions
• Valuations on the space of quasi-concave functions
• Localization of valuations and Alesker's irreducibility theorem