Abstract. We say that a function from X = C L [0, 1] is k-convex (for k ≤ L) if its kth derivative is nonnegative. Let P denote a projection from X onto V = Π n ⊂ X, where Π n denotes the space of algebraic polynomials of degree less than or 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 = (σ 0 , σ 1 , . . . , σ n ) be an (n + 1)-tuple with σ i ∈ {0, 1}; we say f ∈ X is multi-convex if f (i) ≥ 0 for i such that σ i = 1. 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 C L [0, 1].1. Introduction. When X is a Banach space and V ⊂ X a subspace, we denote by P(X, V ) the set of all projections from X onto V ; in the cases where there is no ambiguity, we will simply write P. We say that a projection P 0 is minimal if P 0 ≤ P for all P ∈ P(X, V ).There exist a large number of papers concerning minimal projections. The problems considered are mainly existence ([15] While a minimal projection will, in general, provide good approximations, it may fail to preserve particular properties of elements, as illustrated below. We are therefore motivated to look for projections which leave invari-