  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
$p>n-2$.
