The Hausdorff dimension of sets containing circles in many directions

TL;DR

This paper investigates the Hausdorff dimension of sets in Euclidean space that contain all meridians of a sphere, establishing a lower bound of n-1 for their dimension.

Contribution

It proves that any set containing all meridians of a sphere in br^n must have Hausdorff dimension at least n-1, providing a new geometric dimension bound.

Findings

Sets containing all meridians have Hausdorff dimension br^{n-1} or higher.

The dimension bound applies to sets with specific geometric structures.

The result extends understanding of the geometric complexity of such sets.

Abstract

Let us consider a sphere $S^{n-1}$ of radius $r$ in $\mathbb{R}^n$, where we have fixed poles $N$ and $S$. Suppose that $K$ is a set in $\mathbb{R}^n$ containing a translated copy of each meridian (that is an $S^{n-2}$-sphere) of $S^{n-1}$. Then the Hausdorff dimension of $K$ must be bigger than or equal to $n-1$.