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 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).
