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Anisotropic Minkowski Content for Countably $\mathcal{H}^k$-rectifiable Sets

Filip Fryš

TL;DR

This work extends Federer’s isotropic, lower-dimensional Minkowski content to anisotropic settings, showing that for a $k$-rectifiable compact set $S$ and any convex body $C$, the $C$-anisotropic $k$-Minkowski content exists and equals $\mathcal{M}^k_{C}(S)=\frac{1}{\omega_{n-k}}\int_S \mathcal{H}^{n-k}(C_x)\,d\mathcal{H}^k(x)$ with $C_x = P_{(\textup{T}^k_x S)^{\perp}}(C)$. Under a suitable AFP-condition (and its relative-$L$ version), this result extends to countably $\mathcal{H}^k$-rectifiable compact sets, with the same integral representation, while the case balance $k+m-n$ dictates whether a positive lower bound, an exact integral form, or a zero limit holds. The paper also proves that, once the content exists for one full-dimensional $C$, it exists and yields the same value for all full-dimensional convex bodies. These results generalize classical isotropic findings to anisotropic, lower-dimensional contexts, enriching the geometric measure theory toolkit for analyzing anisotropic volumes and their dependence on the structuring body $C$.

Abstract

This paper investigates the existence of the anisotropic lower-dimensional Minkowski content. We establish that the $C$-anisotropic $k$-dimensional Minkowski content of a $k$-rectifiable compact set always exists and coincides with a specific functional that depends naturally on $C$. We further show that the same conclusion holds for countably $\mathcal{H}^k$-rectifiable compact sets, provided that the so-called \emph{AFP-condition} is satisfied. In addition, we discuss how the existence of the $C$-anisotropic $k$-dimensional Minkowski content for a countably $\mathcal{H}^k$-rectifiable compact set depends on the choice of $C$.

Anisotropic Minkowski Content for Countably $\mathcal{H}^k$-rectifiable Sets

TL;DR

This work extends Federer’s isotropic, lower-dimensional Minkowski content to anisotropic settings, showing that for a -rectifiable compact set and any convex body , the -anisotropic -Minkowski content exists and equals with . Under a suitable AFP-condition (and its relative- version), this result extends to countably -rectifiable compact sets, with the same integral representation, while the case balance dictates whether a positive lower bound, an exact integral form, or a zero limit holds. The paper also proves that, once the content exists for one full-dimensional , it exists and yields the same value for all full-dimensional convex bodies. These results generalize classical isotropic findings to anisotropic, lower-dimensional contexts, enriching the geometric measure theory toolkit for analyzing anisotropic volumes and their dependence on the structuring body .

Abstract

This paper investigates the existence of the anisotropic lower-dimensional Minkowski content. We establish that the -anisotropic -dimensional Minkowski content of a -rectifiable compact set always exists and coincides with a specific functional that depends naturally on . We further show that the same conclusion holds for countably -rectifiable compact sets, provided that the so-called \emph{AFP-condition} is satisfied. In addition, we discuss how the existence of the -anisotropic -dimensional Minkowski content for a countably -rectifiable compact set depends on the choice of .
Paper Structure (12 sections, 27 theorems, 249 equations)

This paper contains 12 sections, 27 theorems, 249 equations.

Table of Contents

  1. Preliminaries
  2. Notation
  3. Geometric Measure Theory
  4. Convex Bodies
  5. Anisotropic $k$-dimensional Minkowski Content for Convex Bodies and Compact $\mathcal{C}^1$ $k$-graphs
  6. Anisotropic $k$-dimensional Minkowski Content for Countably $\mathcal{H}^k$-rectifiable Compact Sets
  7. Lower Bound for General Countably $\mathcal{H}^k$-rectifiable Sets
  8. The Family of AFP-conditions
  9. The Case $k+m-n>0$
  10. The Case $k+m-n=0$
  11. The Case $k+m-n<0$
  12. Dependence on the Choice of $C$

Key Result

Theorem 1.2

Let $f\colon\mathbb{R}^k\to\mathbb{R}^n$ be Lipschitz and $E\subseteq\mathbb{R}^k$ be $\mathcal{H}^k$-measurable, where $k, n\in\mathbb{N}$ satisfy $1\leq k\leq n$. Then the map $y\mapsto \mathcal{H}^{0}(f^{-1}(\{y\})\cap E)$ is $\mathcal{H}^k$-measurable and

Theorems & Definitions (63)

  • Definition 1.1
  • Theorem 1.2: Area Formula, federer
  • Theorem 1.3: Coarea Formula, federer
  • Definition 1.4
  • Theorem 1.5: federer
  • Remark 1.6: zahle
  • Definition 1.7
  • Remark 1.8: ambrosio
  • Theorem 1.9: ambrosio
  • Remark 1.10: ambrosio
  • ...and 53 more