Invariant Valuations on Star-Shaped Sets

Abstract

The Brunn Minkowski theory of convex bodies and mixed volumes has provided many tools for solving problems involving projections and valuations of compact convex sets in Euclidean space. Among the most beautiful results of twentieth century convexity is Hadwiger's characterization theorem for the elementary mixed volumes (Quermassintegrals); (see [3, 5, 9]). Hadwiger's characterization leads to effortless proofs of numerous results in geometric convexity, including mean projection formulas for convex bodies [13, p. 294] and various kinematic formulas [7, 12, 14, 15]. Hadwiger's theorem also provides a connection between rigid motion invariant set functions and symmetric polynomials [1, 7]. Recently, advancements have been made in a theory introduced by Lutwak [8] that is dual to the Brunn Minkowski theory, a theory tailored for dealing with analogous questions involving star-shaped sets and intersections with subspaces (see also [2, 4, 6]). In the dual theory convex bodies are replaced by star-shaped sets, and support functions are replaced by radial functions. Hadwiger's characterizaton theorem is of such fundamental importance that any candidate for a dual theory must possess a dual analogue. However, the dual theory in its original form was not sufficiently rich to be able to accommodate a dual of Hadwiger's theorem. In [6], it was shown that the natural setting for the dual theory is larger than that envisioned by previous investigators. By defining the dual topology on star-shaped sets in terms of the L topology on the space of n-integrable functions on the unit sphere, the author was able to extend the dual theory to a broad class of star-shaped sets, called L-stars. Many new theorems can be proved within this larger framework, including a Hadwiger-style classification theorem for continuous valuations on star-shaped sets that are homogeneous with respect to dilation. In the present paper we discard the stringent requirement of homogeneity and continue with classification theorems for continuous valuations on article no. AI971601