Polynomial approximation on convex subsets of $\mathbb R^n

Abstract

Let K be a closed bounded convex subset of $\Bbb R^n$; then by a result of the first author, which extends a classical theorem of Whitney there is a constant $w_m(K)$ so that for every continuous function f on K there is a polynomial $ϕ$ of degree at most m-1 so that $$ |f(x)-ϕ(x)|\le w_m(K)\sup_{x,x+mh\in K} |Δ_h^m(f;x)|.$$ The aim of this paper is to study the constant $w_m(K)$ in terms of the dimension n and the geometry of K. For example we show that $w_2(K)\le \frac12[\log_2n]+\frac54$ and that for suitable K this bound is almost attained. We place special emphasis on the case when K is symmetric and so can be identified as the unit ball of finite-dimensional Banach space; then there are connections between the behavior of $w_m(K)$ and the geometry (particularly the Rademacher type) of the underlying Banach space. It is shown for example that if K is an ellipsoid then $w_2(K)$ is bounded, independent of dimension, and $w_3(K)\sim \log n.$ We also give estimates for $w_2$ and $w_3$ for the unit ball of the spaces $\ell_p^n$ where $1\le p\le \infty.$