Polynomial Approximation on Convex Subsets of Rn

Abstract. Let K be a closed bounded convex subset of Rn ; then by a result of the first author, which extends a classical theorem of Whitney there is a constantwm(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)| ≤ wm(K ) sup x,x+mh∈K |1h ( f ; x)|. The aim of this paper is to study the constant wm(K ) in terms of the dimension n and the geometry of K . For example, we show that w2(K ) ≤ 2 [log2 n] + 4 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 wm(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 w2(K ) is bounded, independent of dimension, andw3(K ) ∼ log n. We also give estimates forw2 andw3 for the unit ball of the spaces `p where 1 ≤ p ≤ ∞.