Invariant Valuations on Super-Coercive Convex Functions

Abstract

AbstractAll non-negative, continuous,$\text{SL}(n)$, 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,$\text{SL}(n)$, and dually translation invariant valuations is obtained. In both cases, different functional analogs of the Euler characteristic, volume, and polar volume are characterized.