The Legendre transform, the Laplace transform and valuations

Abstract

We first prove that the Legendre transform is the only continuous and $\mathrm{SL}(n)$ contravariant valuation that behaves as a conjugation of two important translations on super-coercive, lower semi-continuous, and convex functions. Then we turn to a similar setting on log-concave functions and find characterizations of not merely the duality transform but also the Laplace transform on log-concave functions. With the notion of dual valuation, we also obtain characterizations of the identity transform on finite convex functions and positive log-concave functions.

References in corpus (17)

• Crofton measures and Minkowski valuations
• Valuations on Convex Functions
• Minkowski valuations on convex functions
• A homogeneous decomposition theorem for valuations on convex functions
• Laplace transforms and valuations
• The support of dually epi-translation invariant valuations on convex functions
• Volume, Polar Volume and Euler Characteristic for Convex Functions
• Hessian valuations
• The Hadwiger theorem on convex functions, IV: The Klain approach
• The Hadwiger theorem on convex functions, III: Steiner formulas and mixed Monge-Ampère measures
• Real-valued valuations on Sobolev spaces
• The Hadwiger theorem on convex functions, I
• $\mathrm{SL}(n)$ invariant valuations on super-coercive convex functions
• The Hadwiger theorem on convex functions, II: Cauchy-Kubota formulas
• $\mathrm{SL}(\mathrm{n})$ covariant function-valued valuations
• Equivariant Endomorphisms of Convex Functions
• Geometric valuation theory