A classification of SL(n) invariant valuations

Abstract

A classification of upper semicontinuous and SL(n) invariant valuations on the space of n-dimensional convex bodies is established. As a consequence, complete characterizations of centro-affine and Lp affine surface areas are obtained. The proofs make use of a new SL(n) shaping process for convex bodies. In his 1900 ICM Address, David Hilbert asked in his Third Problem whether an elementary definition for volume of polytopes is possible. Max Dehn's solution in 1901 makes critical use of the notion of valuations, that is, of functions Φ: S{script} → R{double-struck} that satisfy the inclusion-exclusion relation. whenever K, L, K ∪ L, K ∩ L ∈ S, where S is a collection of sets. Dehn solved Hilbert's Third Problem by constructing a rigid motion invariant valuation which vanishes on lower dimensional sets and is not equal to volume (under any normalization). Since then investigations of valuations have been an active and prominent part of mathematics (see [1]-[8], [16], [19], [20], [24], [25], [31]-[34], and [53] for some of the more recent results). Dehn's work has been strengthened considerably by Sydler and Hadwiger. A systematic study of valuations was initiated by Hadwiger, who was in particular interested in classifying valuations on the set, Kn, of convex bodies (compact convex sets) in Rn. Probably the most famous result on valuations is the Hadwiger characterization theorem. © 2010 by Princeton University.