Some approximation problems in semi-algebraic geometry

TL;DR

This paper investigates approximation problems in semi-algebraic geometry, establishing semi-algebraicity of certain sets, analyzing critical points, and introducing a generalized distance degree for various norms.

Contribution

It introduces the concept of the $ u$- distance degree and extends approximation theory to semi-algebraic sets with different norms, including the $ ext{l}^p$ norm.

Findings

Semi-algebraicity of the set of points with unique approximation

Introduction of the $ u$- distance degree for algebraic sets

Analysis of approximation with $ ext{l}^p$ norms

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 appearing in \cite{DHOST} for the Euclidean norm. We discuss separately the case of the $\ell^p$ norm ($p>1$).