Abstract
A geometric framework relating valuations on convex bodies to valuations on convex functions is introduced. It is shown that a classical result by McMullen can be used to obtain a characterization of continuous, epi-translation invariant, and n-epi-homogeneous valuations on convex functions, which was previously established by Colesanti, Ludwig, and Mussnig. Following an approach by Goodey and Weil, a new characterization of 1-epi-homogeneous valuations is obtained.
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Acknowledgements
The authors would like to thank Monika Ludwig and Fabian Mussnig for many insightful comments and discussions, and the anonimus referee for thorough and constructive comments greatly improving the presentation. The second named author was supported, in part, by the Austrian Science Fund (FWF): 10.55776/P34446, and by the Gruppo Nazionale per l’Analisi Matematica, la Probabilitá e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
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