Problème inverse du lieu des meilleures approximations linéaires (Q1083635)

Let E be a normed space and V a finite-dimensional subspace of E. Given \(f\in E\setminus V\), the set of best approximation to f from V is nonempty, convex and compact: it is the solution set of the problem inf\(\{\Phi\) (y,-1); \(y\in R^ n\}\), where \(\Phi (x_ 1,...,x_ n,\quad x_{n+1})=\| x_ 1f_ 1+...x_ nf_ n+x_{n+1}f\|\) and \(f_ 1,...,f_ n\) is a basis of V. The following result is proved here: given a compact, convex subset K of \(R^ n\), \(n+1\) linearly independent vectors \(f_ 1,...,f_ n\), f exist in C([a,b],R) (or also, in any universal normed space) such that K is the set of best approximations to f from \(V=[f_ 1,...,f_ n].\) Several remarks concerning the possibility of generating a norm in \(R^ n\) by the norm in a space E are made; these and other related results discussed in the paper are taken from the author's thesis [IMAG, Univ. de Grenoble (1972)].