The Mean Width of a Convex Polytope

Abstract

1. Let S n ~ l be the unit ( « — l)-sphere centred on the origin in Euclidean space E". For each unit vector weS" " 1, the width w(K, u) of a closed bounded convex set K in direction u is the distance between the supporting hyperplanes of K whose outward normals are u and — w. The mean width mn{K) is then defined by, (1) where /.i is the Lebesgue measure on S " \\ Many properties of the function mn(K) are well known; see, for example, [2; p. 211]. In this paper we prove a new identity connecting the mean widths of a polytope and its faces: (2) THEOREM. Let P be any convex d-dimensional polytope in E " (n^d), and, for O^j^d—l, let F / (i=\\,-ifjiP)) represent its j-faces. Then the following relation holds: (1 + (- l) d) mn{P) = 'gmtft 1)- jT m.iF?) +... + (- l) d ''g ™ mn{F("'). (3) In the case n=d = 2, because of Cauchy's surface-area formula [2; p. 208], relation (3) reduces to the trivial statement that the perimeter of a convex polygon is equal to the sum of the lengths of its edges. If K is any closed bounded convex set of dimension at most d, we may write KaE d czE n and use md(K) and mn(K) to denote the mean widths of K relative to the spaces E d and E n respectively. A simple calculation shows that for all such K, mn(K) is a constant multiple of md(K), to be precise, mn(K) md{K) where B is the beta function. We deduce that if (3) holds for n=d, then it will hold for all n^d, and therefore it will suffice to prove the theorem in the case n=d only. Because of this fact, it is of interest to convert (3) into a form which is independent of the dimension of the space in which P is embedded. This may be done in the following manner. Define a function M(K), which may be called the absolute mean width, by where n is any integer satisfying w^dimiC, and KaE". Then it is easily verified that the value of M(K) depends only on K and not on n, and since M(K) is a constant multiple of mn(K), the relation analogous to (3), obtained by replacing mn by M throughout, will hold.