Summary: 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 \(K:=\mathrm{Im}(\phi)\).
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 \(\alpha>\frac{(n-2)n}{(n-1)^2}\),

(2)

\(c\) is continuous and \(c\vert_{S^{n-2}}\in W^{1,p}(S^{n-2})\) for some \(p>n-2\).