On Kakeya maps with regularity assumptions

TL;DR

This paper investigates Kakeya sets parametrized by Kakeya maps in n, showing they have positive measure under certain regularity conditions on the defining functions, extending understanding of geometric measure properties.

Contribution

It establishes conditions under which Kakeya sets have positive measure, specifically when the defining map's boundary function has certain regularity properties.

Findings

Kakeya sets have positive measure if the boundary function is Hölder continuous with lpha>rac{(n-2)n}{(n-1)^2}.

Kakeya sets have positive measure if the boundary function belongs to the Sobolev space W^{1,p} with p>n-2.

The results connect regularity assumptions on boundary functions to measure-theoretic properties of Kakeya sets.

Abstract

In $\mathbb R^n$, we parametrize Kakeya sets using Kakeya maps. A Kakeya map is defined to be a map $$\phi:B^{n-1}(0,1)\times [0,1]\rightarrow \mathbb{R}^{n}, (v,t)\mapsto (c(v)+tv,t),$$ where $ c:B^{n-1}(0,1)\rightarrow \mathbb{R}^{n-1}$. The associated Kakeya set is defined to be $ K:=\text{Im} (\phi). $ We show that the Kakeya set $K$ has positive measure if either one of the following conditions is true. (1) $c$ is continuous and $c|_{S^{n-2}}\in C^\alpha(S^{n-2})$ for some $\alpha>\frac{(n-2)n}{(n-1)^2}$, (2) $c$ is continuous and $c|_{S^{n-2}}\in W^{1,p}(S^{n-2})$ for some $p>n-2$.