Minkowski and packing Dimension comparisons for sets with Reifenberg properties

TL;DR

This paper explores the relationships between Minkowski and packing dimensions for sets with Reifenberg properties, extending previous classifications based on Hausdorff measures to include fractal dimensions and measure equivalences.

Contribution

It extends the classification of Reifenberg properties by incorporating Minkowski and packing dimensions, and investigates measure and rectifiability aspects.

Findings

Established connections between Minkowski and packing dimensions for Reifenberg sets.

Extended classification to include measure and rectifiability properties.

Identified conditions for equality of packing and Hausdorff measures.

Abstract

In Koeller \cite{koerprops} the twelve variants of the Reifenberg properties known to be instrumental in the theory of minimal surfaces were classified with respect to various Hausdorff measure based measure theoretic properties. The classification lead to the consideration of fine geometric properties and a connection to fractal geometry. The current work develops this connection and extends the classification to consider Minkowski-dimension, packing dimension, measure, and rectifiability, and the equality of packing and Hausdorff measures with interesting results.