Some means of convex bodies

Abstract

The systematic study of various mean-values of nonnegative valued functions is due in large part to the work of J. L. W. V. Jensen [12], A. Kolmogoroff [11], and B. Jessen [13]. In this paper we present a similar theory of certain means of functions whose values are taken from a collection of convex bodies in a finite dimensional Euclidean space. The range of nonnegative valued functions is totally ordered while the collection of convex bodies to be considered is partially ordered by set inclusion; corresponding to inequalities, we shall have inclusions, here to be called inequalities, between mean-values. Thus the discussion will furnish an example of a partially ordered system whose algebraic and topological structure is sufficient to admit a fairly elaborate theory of inequalities analogous to that of the real numbers. The first section is a resume of those parts of the work of Jensen, Kolmogoroff, and Jessen for which analogues will be developed. ?2 treats of pertinent material about star and convex bodies. Certain families of star bodies and of convex bodies are defined in the third section; these latter play the role of nonnegative valued functions. Two systems of power means of such families are described. Each system is, in its way, anialogous to the power means of nonnegative valued functions. In ?4 we discuss some special cases of power means including elementary means and the Riemaiin-Minkowski integrals of A. Dinghas [2], [3]. Also some rotation invariants of a convex body are described as power means of special families determined by the convex body. The fifth section begins with crucial properties of the means and contains an extremal characterization for the two systems of means defined in the third section. Analogues of Jensen's and Jessen's inequalities make up ?6. We also mentioni the limiting cases of power means as the power index becomes infinite, positively and negatively. As an applicationl of Jessen's inequality, a Brunn-Minkowski type theorem is proved. In the final section we discuss some further systems of meanls of convex bodies. 1. The definition of the elementary power means extends in a natural way to include power means of certain functions f which have finite, nonnegative values and whose domain T is a compact topological group equipped with its Haar measure. Iff is bounded and measurable over its domain, its pth- power mean is defined by