Geometrical Characterization of Strict Suns in $\ell^\infty(n)$

A subset $M$ of a normed linear space $X$ is called a strict sun if, for any $x\in X\setminus M$, the set of its nearest points from $M$ is nonempty and for any point $y\in M$ which is nearest to $x$, the point $y$ is a nearest point from $M$ to any point of the ray $\{\lambda x+(1-\lambda)y\mid\lambda>0\}$. We give an intrinsic geometrical characterization of strict suns in the space $\ell^\infty(n)$.