Preservation of approximative properties of subsets of Chebyshev sets and suns in $\ell^\infty (n)$

We describe the subsets $\Pi\subset\mathbb R^n$ whose intersection with a Chebyshev set (sun, or strict sun) $M$ in $\ell^\infty(n)$ preserves the approximative properties of $M$ in $\ell^\infty(n)$.