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arXiv · 2402.14731

Lefschetz operators on convex valuations

Leo Brauner·Georg C. Hofstätter·Oscar Ortega-Moreno
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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.

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Leo Brauner, Georg C. Hofstätter, Oscar Ortega-Moreno. 2024-02-22. Lefschetz operators on convex valuations. https://arxiv.org/abs/2402.14731

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