Intersection Bodies and Valuations

All GL (n) covariant star-body-valued valuations on convex polytopes are completely classified. It is shown that there is a unique nontrivial such valuation. This valuation turns out to be the so-called “intersection operator”—an operator that played a critical role in the solution of the Busemann-Petty problem. Introduction. A function Z defined on the set K of convex bodies (that is, of convex compact sets) in Rn or on a certain subset C of K and taking values in an abelian semigroup is called a valuation if Z K + Z L = Z (K ∪ L) + Z (K ∩ L), whenever K, L, K∪L, K∩L ∈ C. Real valued valuations are classical and Blaschke obtained the first classification of such valuations that are SL (n) invariant in the 1930s. This was greatly extended by Hadwiger in his famous classification of continuous, rigid motion invariant valuations and characterization of elementary mixed volumes. See [13], [17], [32], [33] for information on the classical theory and [1]–[4], [15], [16], [25], [26], [28] for some of the more recent results. In [24], [27], a classification of convex-body-valued valuations Z: P → K was obtained where P is the set of convex polytopes in Rn containing the origin and addition in K is Minkowski addition of convex bodies (defined by K + L = {x + y: x ∈ K, y ∈ L}). A valuation Z is called GL (n) covariant, if there exists a q ∈ R such that for all φ ∈ GL (n) and all bodies K, Z (φK) = |detφ| φZ K, where detφ is the determinant of φ. It is called GL (n) contravariant, if there exists a q ∈ R such that φ ∈ GL (n) and all bodies K Z (φK) = |detφ| φ−t Z K, Manuscript received December 3, 2004; revised October 4, 2005. American Journal of Mathematics 128 (2006), 0–00. 1