Abstract Steiner Points for Convex Polytopes

STEINER POINTS FOR CONVEX POLYTOPES CHRISTIAN BERG Let & denote the set of all convex polytopes, degenerate or not, in ^-dimensional Euclidean space E. An abstract Steiner point for convex polytopes in E is a mapping S:2P-+E satisfying S(P+Q) = S(P) + S(Q) for all P, Qe0*, (1) addition on the left being Minkowski addition of convex sets, and S{a(P)) = o{S(P)) (2) for all Pe&* and all similarity transformations a in E. (A similarity means any composition of dilatations and isometries.) In the present paper we shall characterize the abstract Steiner points for convex polytopes in two and three dimensions. As a simple consequence of this characterization we find that an abstract Steiner point S for convex polytopes in E, d = 2,3, satisfying S(P) e P for all P e &, is the usual Steiner point. For the definition and properties of this point see [3]. For any vertex A of Pe^} let V(A) denote the closed convex spherical polytope in S~, consisting of all outer normal directions for supporting hypeiplanes of P through A. The external angle u at the vertex A is now defined as u = o)(V(A)), where a> denotes the normalized surface measure of S~. Let \i be any mapping from the set "T~ of closed convex spherical polytopes in S~ to the real numbers satisfying (3) whenever V, W, V u Wcf', and the polytopes V and W are without common interior points, n(6(V)) = n(V), (4) for all isometries 8 of S", and niS"-) = 1. (5) The set of all these mappings /J will be denoted by M. For any /jeMwe define an abstract Steiner point S^ by the formula SP(P) = Z ^(K04,))pf, (6) where pt denotes the position vector of the vertex Ah (i = 1,..., n) of Pe^ . For the proof of (1) and (2) for Ŝ see [ 3; p. 1296]. Examples of such mappings \i are obtained as follows. Let O denote the set of functions (j>: [0,1] -* R satisfying (u + v) = 4>{u)+(v) (7) Received 26 February, 1970; revised 30 October, 1970. [J. LONDON MATH. SOC. (2), 4 (1971), 176-180] ABSTRACT STEINER POINTS FOR CONVEX POLYTOPES 177STEINER POINTS FOR CONVEX POLYTOPES 177 whenever w, v, u + ve [0,1], and