# $\operatorname{SL}(n)$ invariant valuations on super-coercive convex functions

## TL;DR

This paper classifies all non-negative, continuous, SL(n) and translation invariant valuations on super-coercive convex functions, and also those invariant under the Legendre transform, identifying analogs of classical geometric invariants.

## Contribution

It provides a complete classification of such valuations on super-coercive convex functions, including dual invariance under the Legendre transform, extending geometric valuation theory.

## Key findings

- Characterization of valuations analogous to Euler characteristic, volume, and polar volume.
- Classification of SL(n) and translation invariant valuations on super-coercive convex functions.
- Extension to dual invariance under Legendre transform.

## Abstract

All non-negative, continuous, $\operatorname{SL}(n)$ and translation invariant valuations on the space of super-coercive, convex functions on $\mathbb{R}^n$ are classified. Furthermore, using the invariance of the function space under the Legendre transform, a classification of non-negative, continuous, $\operatorname{SL}(n)$ and dually translation invariant valuations is obtained. In both cases, different functional analogs of the Euler characteristic, volume and polar volume are characterized.