Convex polynomial and spline approximation in L_p, 0 < p < ∞

We prove that a convex function f ∈ L_p[-1, 1], 0 < p < ∞, can be approximated by convex polynomials with an error not exceeding C ω^φ₃ (f, 1/n)_p where ω^φ₃ (f, ·) is the Ditzian-Totik modulus of smoothness of order three of f. We are thus filling the gap between previously known estimates involving ω^φ₂ (f, 1/n)_p, and the impossibility of having such estimates involving ω₄. We also give similar estimates for the approximation of f by convex C⁰ and C¹ piecewise quadratics as well as convex C² piecewise cubic polynomials.