We say that a function from X = CL[a, b] is k-convex (for k ≤ L) if the kth derivative of the function is nonnegative. Let P denote a projection from X onto V = Πn ⊂ X, where Πn denotes the space of algebraic polynomials of degree less than equal to n. If we want P to leave invariant the cone of k-convex functions (k ≤ n), we find that such a demand is impossible to fulfill for nearly every k. Indeed only for k = n−1 and k = n does such a projection exist. So let us consider instead a more general ’shape’ to preserve. Let σ = (σ0, σ1, . . . , σn) be an (n + 1)-tuple with σi ∈ {0, 1}; we say f ∈ X is multi-convex if f (i) ≥ 0 for i such that σi = 1. In this paper we characterize those σ for which there exists a projection onto V preserving the multi-convex shape. For those shapes able to be preserved via a projection, we construct (in all but one case) a minimal norm multi-convex preserving projection. Out of necessity, we include some results concerning the geometrical structure of CL[a, b] Department of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Krakow, Poland Department of Mathematics, University of Northern Iowa, Cedar Falls Iowa, USA