Monotone Valuations on the Space of Convex Functions

Abstract

We consider the space Cn of convex functions u defined in Rn with values in R ∪ {∞}, which are lower semi-continuous and such that lim|x| } ∞ u(x) = ∞. We study the valuations defined on Cn which are invariant under the composition with rigid motions, monotone and verify a certain type of continuity. We prove integral representations formulas for such valuations which are, in addition, simple or homogeneous.

Keywords

• convex functions
• valuations
• convex bodies
• sub-level sets
• intrinsic volumes
• 26b25; 52a41; 52b45