Monotone valuations on the space of convex functions

Abstract

We consider the space of convex functions defined in the Euclidean $n$-dimensional space, which are lower semi-continuous and tend to infinity at infinity. We study real-valued valuations defined on this space of functions, which are invariant under the composition with rigid motions, monotone and verify a certain type of continuity. Among these valuations we prove integral representation formulas for those which are, additionally, simple or homogeneous.

55 page, 4 figures

Cited by in corpus (13)

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• A homogeneous decomposition theorem for valuations on convex functions
• Laplace transforms and valuations
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• Real-valued valuations on Sobolev spaces
• The Hadwiger theorem on convex functions, I
• Equivariant Endomorphisms of Convex Functions
• Metrics and Isometries for Convex Functions
• Geometric valuation theory
• Non-trivial translation-invariant valuations on $L^{\infty }$
• Dot product invariant valuations on Lip$(S^{n-1})$