In this paper we deal with a best approximation of a vector with respect to a closed semi-algebraic set C in the space ℝⁿ endowed with a semi-algebraic norm ν. Under additional assumptions on ν 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 ν-distance degree, generalizing the notion developed by other authors for the Euclidean norm. We...

Let $X$ be a finite dimensional Banach space and let $Y\subset X$ be a hyperplane. Let $\text{L}{}_Y=\{L\in \text{L}(X,Y):L{\mid }_Y=0\}$. In this note, we present sufficient and necessary conditions on $L_0\in \text{L}{}_Y$ being a strongly unique best approximation for given $L\in \text{L}\left(X\right)$. Next we apply this characterization to the case of $X=l_{\infty }^n$ and to generalization of Theorem I.1.3 from [12] (see also [13]).