The existence of a homogeneous decomposition for continuous and epi-translation invariant valuations on super-coercive functions is established. Continuous and epi-translation invariant valuations that are epi-homogeneous of degree n are classified. By duality, corresponding results are obtained for valuations on finite-valued convex functions.2000 AMS subject classification: 52B45 (26B25, 52A21, 52A41)

All continuous, SL(n) and translation invariant valuations on the space of convex functions on R n are completely classified. 2000 AMS subject classification: 26B25 (46A40, 52A20, 52B45).

A complete classification of all continuous, epi-translation and rotation invariant valuations on the space of super-coercive convex functions on R n is established. The valuations obtained are functional versions of the classical intrinsic volumes. For their definition, singular Hessian valuations are introduced.

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