$\mathrm{SL}(n)$ invariant valuations on super-coercive convex functions

Abstract

All non-negative, continuous, $\mathrm{SL}(n)$ and translation invariant valuations on the space of super-coercive, convex functions on ${\mathbb{R}}^n$ are classified. Furthermore, using the invariance of the function space under the Legendre transform, a classification of non-negative, continuous, $\mathrm{SL}(n)$ and dually translation invariant valuations is obtained. In both cases, different functional analogs of the Euler characteristic, volume and polar volume are characterized.

References in corpus (4)

• Valuations on Convex Functions
• Minkowski valuations on convex functions
• Symmetric Mahler's conjecture for the volume product in the three dimensional case
• Volume, Polar Volume and Euler Characteristic for Convex Functions

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• The support of dually epi-translation invariant valuations on convex functions
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• The Hadwiger theorem on convex functions, I
• Valuations on Convex Bodies and Functions
• Metrics and Isometries for Convex Functions
• Dot product invariant valuations on Lip$(S^{n-1})$
• A Riesz representation theorem for log-concave functions
• The Legendre transform, the Laplace transform and valuations