  In this paper we deal with a best approximation of a vector with respect to a
closed semi-algebraic set $C$ in the space $\mathbb{R}^n$ endowed with a
semi-algebraic norm $\nu$. Under additional assumptions on $\nu$ we prove
semi-algebraicity of the set of points of unique approximation and other sets
associated with the distance to $C$. For $C$ irreducible algebraic we study the
critical point correspondence and introduce the $\nu$- distance degree,
generalizing the notion appearing in \cite{DHOST} for the Euclidean norm. We
discuss separately the case of the $\ell^p$ norm ($p>1$).
