The Legendre transform, the Laplace transform and valuations

TL;DR

This paper characterizes the Legendre and Laplace transforms as unique valuations with specific invariance properties on convex and log-concave functions, revealing their fundamental roles in convex analysis.

Contribution

It provides the first characterization of the Legendre transform as a unique continuous, SL(n)-contravariant valuation, and extends similar characterizations to the Laplace transform on log-concave functions.

Findings

Legendre transform is uniquely characterized among valuations.

Laplace transform is characterized on log-concave functions.

Dual valuations lead to characterizations of identity transforms.

Abstract

We first prove that the Legendre transform is the only continuous and $\mathrm{SL}(n)$ contravariant valuation that behaves as a conjugation of two important translations on super-coercive, lower semi-continuous, and convex functions. Then we turn to a similar setting on log-concave functions and find characterizations of not merely the duality transform but also the Laplace transform on log-concave functions. With the notion of dual valuation, we also obtain characterizations of the identity transform on finite convex functions and positive log-concave functions.