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-

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Let H n be an n-dimensional Haar subspace of X ¼ C R ½a; b and let H nÀ1 be a Haar subspace of H n of dimension n À 1: In this note we show (Theorem 6) that if the norm of a minimal projection from H n onto H nÀ1 is greater than 1, then this projection is an interpolating projection. This is a surprising result in comparison with Cheney and Morris () which shows that there is no interpolating minimal projection from C½a; b onto the space of polynomials of degree pn; ðnX2Þ: Moreover, this minimal projection is unique (Theorem 9). In particular, Theorem 6 holds for polynomial spaces, generalizing a result of Prophet [(J.

Let X denote a (real) Banach space and V an n-dimensional subspace. We denote by B = B(X, V ) the space of all bounded linear operators from X into V; let P(X, V ) be the set of all projections in B. For a given cone S ⊂ X, we denote by P = P S (X, V ) the set of operators P ∈ P such that P S ⊂ S. When P S = ∅, we characterize those P ∈ P S for which P is minimal. This characterization is then utilized in several applications and examples.

We construct a minimal projection P: X Ä V 3 , where X=[1, t, t 2 , t |t| _ ] and V 3 =[1, t, t 2 ], for all _ 1. This generalizes a result of G. J. O. Jameson. 1996Academic Press, Inc.