Valuations on convex functions and convex sets and Monge–Ampère operators

By Semyon Alesker. Advances in Geometry, 2017

47 citations recorded by the source corpus.

Abstract The notion of a valuation on convex bodies is very classical; valuations on a class of functions have been introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on convex functions which are invariant under adding arbitrary linear functionals, and translation invariant continuous valuations on convex bodies. More precisely, we construct a natural linear map from the former space to the latter and prove that it has dense image and infinite-dimensional kernel. The proof uses the author’s irreducibility theorem and properties of the real Monge–Ampère operators due to A.D. Alexandrov and Z. Blocki. Furthermore we show how to use complex, quaternionic, and octonionic Monge–Ampère operators to construct more examples of continuous valuations on convex functions in an analogous way.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7d8d16a25d09292aef889a34134273ed00238bc2

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

A homogeneous decomposition theorem for valuations on convex functions (2019, background): “As a consequence the polynomiality of epi-translation invariant valuations is obtained and a connection to the valuations introduced by Alesker is established in Section 6.”
The Hadwiger Theorem on Convex Functions, I (2020, result): “Here classification results were obtained for SL(n) invariant valuations in [12, 13, 37, 38], the connection to valuations on convex bodies was explored in [3] and first structural results were established in [14,27,28].”