Smooth valuations on convex functions
- We study continuous dually epi-translation invariant valuations on certain cones of convex functions containing the space of finite-valued convex functions. Using the homogeneous decomposition of this space, we associate a certain distribution to any homogeneous valuation similar to the Goodey-Weil embedding for translation invariant valuations on convex bodies. The support of these distributions induces a corresponding notion of support for the underlying valuations, which imposes certain restrictions on these functionals, and we study the relation between the support of a valuation and its domain. This gives a partial answer to the question which dually epi-translation invariant valuations on finite-valued convex functions can be extended to larger cones of convex functions. We also study topological properties of spaces of valuations with support contained in a fixed compact set. As an application of these results, we introduce the class of smooth valuations on convex functions and show that the subspace of smooth dually epi-translation invariant valuations is dense in the space of continuous dually epi-translation invariant valuation on finite-valued convex functions. These smooth valuations are given by integrating certain smooth differential forms over the graph of the differential of a convex function. We use this construction to give a characterization of a dense subspace of all continuous valuations on finite-valued convex functions that are rotation invariant as well as dually epi-translation invariant. Using results from Alesker's theory of smooth valuations on convex bodies, we also show that any smooth valuation can be written as a convergent sum of mixed Hessian valuations. In particular, mixed Hessian valuations span a dense subspace, which is a version of McMullen’s conjecture for valuations on convex functions.
Author: | Jonas KnörrORCiDGND |
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URN: | urn:nbn:de:hebis:30:3-581211 |
Place of publication: | Frankfurt am Main |
Referee: | Andreas BernigORCiDGND, Raman SanyalORCiDGND, Andrea Colesanti |
Document Type: | Doctoral Thesis |
Language: | English |
Date of Publication (online): | 2021/02/03 |
Year of first Publication: | 2020 |
Publishing Institution: | Universitätsbibliothek Johann Christian Senckenberg |
Granting Institution: | Johann Wolfgang Goethe-Universität |
Date of final exam: | 2020/11/12 |
Release Date: | 2021/02/15 |
Tag: | Integral Geometry; Valuation Theory; Valuation on functions |
Page Number: | 145 |
HeBIS-PPN: | 475792564 |
Institutes: | Informatik und Mathematik / Mathematik |
Dewey Decimal Classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
Sammlungen: | Universitätspublikationen |
Licence (German): | Deutsches Urheberrecht |