Polynomial approximation on convex subsets of Rⁿ

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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 constant wₘ(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),x+mh∈ K |Δₕᵐ(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)≤ 12[₂n]+54 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, and w₃(K) n. We also give estimates for w₂ and w₃ for the unit ball of the spaces ℓₚⁿ where 1≤ p≤ ∞.