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Equivariant Valuations on Convex Functions

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arxiv 2407.08304 v1 pith:M7T264DC submitted 2024-07-11 math.MG math.FA

Georg C. Hofstätter , Jonas Knoerr This is my paper · ORCID
classification math.MGmath.FA
keywords convexfunctionsspacevaluationsbodyfinitevaluesanalogues
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We classify all continuous valuations on the space of finite convex functions with values in the same space which are dually epi-translation-invariant and equi- resp. contravariant with respect to volume-preserving linear maps. We thereby identify the valuation-theoretic functional analogues of the difference body map and show that there does not exist a generalization of the projection body map in this setting. This non-existence result is shown to also hold true for valuations with values in the space of convex functions that are finite in a neighborhood of the origin.

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Cited by 1 Pith paper

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  1. A Paley-Wiener-Schwartz Theorem for smooth valuations on convex functions

    math.FA 2025-05 accept novelty 8.0 of 10

    The paper proves a Paley-Wiener-Schwartz theorem for dually epi-translation invariant valuations on convex functions and uses it to classify all closed affine invariant subspaces.

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