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Let K be a closed bounded convex subset of R; 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) sup x,x+mh∈K |∆h (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 [log2 n] + 5 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, and w3(K) ∼ logn. We also give estimates for w2 and w3 for the unit ball of the spaces l n p where 1 ≤ p ≤ ∞.