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Lefschetz operators on convex valuations

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

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.

Lefschetz operators on convex valuations

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.
Paper Structure (17 sections, 36 theorems, 166 equations)

This paper contains 17 sections, 36 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Mixed spherical projections and liftings
  3. Action on the Klain--Schneider function
  4. Two Radon type transforms
  5. The Lefschetz derivation operator
  6. The Lefschetz integration operator
  7. Minkowski valuations
  8. The Berg functions
  9. The space MVali
  10. Action on the generating function
  11. Spherical valuations
  12. A Legendre type differential equation
  13. Description of rhoi
  14. Klain--Schneider functions of Minkowski valuations
  15. The spherical convolution
  16. ...and 2 more sections

Key Result

Theorem 1

Let $1\leq i\leq n-1$ and $\varphi\in\mathbf{Val}_i$ be smooth and even.

Theorems & Definitions (68)

  • Theorem : Schuster2015
  • Theorem A
  • Theorem B
  • Theorem : Schuster2018Dorrek2017
  • Theorem C
  • Corollary
  • Definition 2.1: Goodey2011
  • Theorem 2.2: Goodey2011
  • Definition 2.3
  • Theorem 2.4: Schneider2014
  • ...and 58 more