The Pólya algorithm on convex sets (Q583534)

For \(x\in R^ n\), define the p-norm of x, \(1\leq p<\infty\), by \(\| x\|_ p=(\sum | x_ i|^ p)^{1/p}\) and define \(\| x\|_{\infty}\), the uniform norm of x, as \(\max | x_ i|\). If M is a closed, convex set in \(R^ n\) and \(y\in R^ n| M\), then \(z\in M\) is said to be (i) best p-norm approximation to y if \(\| z-y\|_ p=\min_{s\in M}\| s-y\|_ p,\) (ii) best \(\ell^{\infty}\)-approximation to y if \(\| z- y\|_{\infty}=\min_{s\in M}\| s-y\|_{\infty},\) Let W be the set of best \(\ell^{\infty}\)-approximations. For each \(z\in W\), let \(| z|\) be the vector whose coordinates are given by the values \(| z_ i|\) arranged in non-decreasing order. Impose the lexicographic ordering on the vector \(| z|\). There exists a unique \(w\in W\) which has \(| w|\) minimal in this ordering. This element is called the strict uniform best approximation. It is shown in this paper (by giving examples) that there exist closed, convex sets in \(R^ n\) for which the best p-norm approximations to a fixed element of \(R^ n\) fail to converge as \(p\to \infty\). Furthermore, it is shown that even if the best approximations converge, they need not converge to the strict best uniform approximation.