A homogeneous decomposition theorem for valuations on convex functions

Abstract

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.

arXiv comment16 pages

References in corpus (10)

• Integral geometry of complex space forms
• Valuations on Convex Functions
• Minkowski valuations on convex functions
• Laplace transforms and valuations
• Monotone valuations on the space of convex functions
• Volume, Polar Volume and Euler Characteristic for Convex Functions
• Hessian valuations
• Real-valued valuations on Sobolev spaces
• $\mathrm{SL}(n)$ invariant valuations on super-coercive convex functions
• Dot product invariant valuations on Lip$(S^{n-1})$

Cited by in corpus (14)

• The support of dually epi-translation invariant valuations on convex functions
• The Hadwiger theorem on convex functions, IV: The Klain approach
• The Hadwiger theorem on convex functions, III: Steiner formulas and mixed Monge-Ampère measures
• The Hadwiger theorem on convex functions, I
• $\mathrm{SL}(\mathrm{n})$ covariant function-valued valuations
• Equivariant Endomorphisms of Convex Functions
• Valuations on Convex Bodies and Functions
• Metrics and Isometries for Convex Functions
• Geometric valuation theory
• A Riesz representation theorem for log-concave functions
• Inequalities and counterexamples for functional intrinsic volumes and beyond
• A Klain-Schneider Theorem for Vector-Valued Valuations on Convex Functions
• On the framework of $L_p$ summations for functions
• The Legendre transform, the Laplace transform and valuations