On Convexity, Mid-Point Convexity and Hausdorff Measures of Sets

TL;DR

This paper characterizes the size of certain Borel sets that are mid-point convex but not convex, using Hausdorff dimensions and measures, providing a complete understanding of their geometric properties.

Contribution

It offers a complete characterization of the Hausdorff measure and dimension of mid-point convex sets that are not convex, filling a gap in geometric measure theory.

Findings

Mid-point convex sets have specific Hausdorff dimension properties.

Such sets can be characterized by their Hausdorff measures.

The results clarify the geometric structure of non-convex mid-point convex sets.

Abstract

We give a complete characterization of the size of Borel sets that are mid-point convex but not (essentially) convex, in terms of their Hausdorff dimensions and Hausdorff measures.