Invariant Valuations on Super-Coercive Convex Functions

All non-negative, continuous, SL(n) , and translation invariant valuations on the space of super-coercive, convex functions on Rⁿ are classified. Furthermore, using the invariance of the function space under the Legendre transform, a classification of non-negative, continuous, 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.