Lefschetz operators on convex valuations
Metric Geometry
2024-02-23 v1
Abstract
We investigate the action of Alesker's Lefschetz operators on translation invariant valuations on convex bodies. For scalar valued valuations, we describe this action on the level of Klain-Schneider functions by a Radon type transform, generalizing a result by Schuster and Wannerer. In the case of rotationally equivariant Minkowski valuations, the Lefschetz operators act on the generating function as a convolution transform. We show that the convolution kernel satisfies a Legendre type differential equation, and thus, is a strictly positive function that is smooth up to one point.
Keywords
Cite
@article{arxiv.2402.14731,
title = {Lefschetz operators on convex valuations},
author = {Leo Brauner and Georg C. Hofstätter and Oscar Ortega-Moreno},
journal= {arXiv preprint arXiv:2402.14731},
year = {2024}
}