2017/03/26 by Semyon Alesker, Alesker, Semyon · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.1703.08778
openalex publication_date 2017/03/26 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
The notion of a valuation on convex bodies is very classical. The notion of a\nvaluation on a class of functions was recently introduced and studied by M.\nLudwig and others. We study an explicit relation between continuous valuations\non convex functions which are invariant under adding arbitrary linear\nfunctionals, and translations invariant continuous valuations on convex bodies.\nMore precisely, we construct a natural linear map from the former space to the\nlatter and prove that it has dense image and infinite dimensional kernel. The\nproof uses the author's irreducibility theorem and few properties of the real\nMonge-Ampere operators due to A.D. Alexandrov and Z. Blocki. Fur- thermore we\nshow how to use complex, quaternionic, and octonionic Monge-Ampere operators to\nconstruct more examples of continuous valuations on convex functions in an\nanalogous way.\n