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The Klain approach to zonal valuations

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

Abstract

We show an analogue of the Klain-Schneider theorem for valuations that are invariant under rotations around a fixed axis, called zonal. Using this, we establish a new integral representation of zonal valuations involving mixed area measures with a disk. In our argument, we introduce an easy way to translate between this representation and the one involving area measures, yielding a shorter proof of a recent characterization by Knoerr. As applications, we obtain various zonal integral geometric formulas, extending results by Hug, Mussnig, and Ulivelli. Finally, we provide a simpler proof of the integral representation of the mean section operators by Goodey and Weil.

The Klain approach to zonal valuations

Abstract

We show an analogue of the Klain-Schneider theorem for valuations that are invariant under rotations around a fixed axis, called zonal. Using this, we establish a new integral representation of zonal valuations involving mixed area measures with a disk. In our argument, we introduce an easy way to translate between this representation and the one involving area measures, yielding a shorter proof of a recent characterization by Knoerr. As applications, we obtain various zonal integral geometric formulas, extending results by Hug, Mussnig, and Ulivelli. Finally, we provide a simpler proof of the integral representation of the mean section operators by Goodey and Weil.

Paper Structure

This paper contains 17 sections, 54 theorems, 141 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Moving between integral representations
  3. Evaluation on cones
  4. Restricting to subspaces
  5. The commuting diagram
  6. Extending from subspaces
  7. The Klain--Schneider theorem for zonal valuations
  8. The three-dimensional case
  9. The induction step
  10. Hadwiger type theorems for zonal valuations
  11. Smooth Valuations
  12. Continuous valuations
  13. Applications
  14. Integral geometric formulas
  15. Mean section operators
  16. ...and 2 more sections

Key Result

Theorem 1.1

For $0 \leq i \leq n$, a valuation $\varphi \in \mathbf{Val}_i(\mathbb{R}^n)$ is rotation invariant if and only if it is a constant multiple of the $i$-th intrinsic volume.

Figures (3)

  • Figure 1:
  • Figure 2: We extend the orthogonal sum $H=E'\oplus F'$, where $P_H e_n\in E'$, to an orthogonal sum $\mathbb{R}^n=E\oplus F'$.
  • Figure 3: The Minkowski sum $\lambda C_s + \mu C_t$.

Theorems & Definitions (98)

  • Theorem 1.1: Hadwiger1957
  • Theorem 1.2: Knoerr2024
  • Theorem A
  • Theorem 1.3: Klain1995Schneider1996
  • Corollary 1.4
  • Theorem B
  • Corollary C
  • Theorem D
  • Theorem E
  • Corollary F
  • ...and 88 more