On Kakeya maps with regularity assumptions

Abstract

In ${\mathbb{R}}^n$, we parametrize Kakeya sets using Kakeya maps. A Kakeya map is defined to be a map $$\varphi :B^{n-1}(0,1)\times [0,1]\to {\mathbb{R}}^n,(v,t)\mapsto (c(v)+tv,t),$$ where $c:B^{n-1}(0,1)\to {\mathbb{R}}^{n-1}$. The associated Kakeya set is defined to be $K:=\text{Im}(\varphi ).$ 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 $p>n-2$.

Some corrections are made; accepted by Math Research Letters