On Kakeya maps with regularity assumptions

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

We show that the Kakeya set $K$ has positive measure if either one of the following conditions is true:

(1) $c$ is continuous and ${c \vert}_{S{n-2}} \in C^\alpha (S{n-2})$ for some $\alpha \gt \frac{(n-2)n}{(n-1)^2}$,

(2) $c$ is continuous and ${c \vert}_{S{n-2}} \in W^{1,p} (S^{n-2})$ for some $p \gt n-2$.

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Received 9 June 2021

Received revised 10 October 2022

Accepted 16 November 2022

Published 21 June 2023