Equivariant Endomorphisms of Convex Functions

TL;DR

This paper characterizes all continuous, additive, and equivariant endomorphisms of various spaces of convex functions, providing a comprehensive understanding of their structure under different symmetry and regularity conditions.

Contribution

It offers a complete classification of equivariant and monotone endomorphisms of convex function spaces, including one-dimensional cases, under continuity and additivity assumptions.

Findings

Characterization of all continuous, additive, GL(n)-equivariant endomorphisms of convex functions.

Classification of monotone, rotation, and dilation-equivariant endomorphisms.

Complete description of endomorphisms in the one-variable convex function space.

Abstract

Characterizations of all continuous, additive and $\mathrm{GL}(n)$-equivariant endomorphisms of the space of convex functions on a Euclidean space $\mathbb{R}^n$, of the subspace of convex functions that are finite in a neighborhood of the origin, and of finite convex functions are established. Moreover, all continuous, additive, monotone endomorphisms of the same spaces, which are equivariant with respect to rotations and dilations, are characterized. Finally, all continuous, additive endomorphisms of the space of finite convex functions of one variable are characterized.