Some approximation problems in semi-algebraic geometry
Volume 107 / 2015
Banach Center Publications 107 (2015), 133-147
MSC: 14P10, 41A52, 41A65.
DOI: 10.4064/bc107-0-9
Abstract
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 developed by other authors for the Euclidean norm. We discuss separately the case of the $\ell^p$ norm ($p \gt 1$).
