Continuity and convexity of projections and barycentric coordinates in convex polyhedra

If sQ, , sn are linearly independent points of real ^-dimensional Euclidean space R then each point x of their convex hull S has a (unique) representation x = Σ?=oM)< with \(%) ^ 0 (i = 0, , n) and Σ?=<Λ<(#) — 1> and the barycentric coordinates λ0, , Xn are continuous convex functions on S (cf. [3, p. 288]). We shall show in this paper that given any finite set 809 •• ,s f n of points of R n we can assign barycentric coordinates λ>o> , λTO to their convex hull S in such a way that each coordinate is continuous on S and that one prescribed coordinate (λ0 say) is convex on £ (Theorem 2); the author does not know whether it is always possible to make all the coordinates convex simultaneously (cf. Example 3). In proving Theorem 2 we shall use certain "projections" which we now define; these projections are in general distinct from those of [1, p. 614] and [2, p. 12], Given two distinct points s0 and s of R , let sos be the open half-line consisting of all points sQ + X(s — s0) with λ > 0; given a point s0 of R n and a closed subset S of R such that s0 $ S, let C(s0, S) be the "cone" formed by the union of all open half-lines sos with s in S; and given a point x in such a cone C(sQ, S), let π(x) be the (unique) point of sox Π S which is closest to s0. Then we shall call the function π the "projection of C(so,S) on S." Our proof of Theorem 2 depends on the fact that if S is a convex polyhedron then π is continuous (Theorem 1). This result may appear to be obvious, but it is not immediately obvious how a formal proof should be given; moreover, as we shall show (Examples 1 and 2), the conclusion need not remain true for polyhedra S which are not convex or for convex sets S which are not polyhedra. The author is indebted to the referee for improvements to Lemma 3, Example 1, and Example 2, and for the remark at the end of § 1.