Even Valuations on Convex Bodies

The notion of even valuation is introduced as a natural generalization of volume on compact convex subsets of Euclidean space. A recent characterization theorem for volume leads in turn to a connection between even valuations on compact convex sets and continuous functions on Grassmannians. This connection can be described in part using generating distributions for symmetric compact convex sets. We also explore some consequences of these characterization results in convex and integral geometry. Recent interest in volume as a valuation on compact convex sets stems from Hilbert’s Third Problem, which is actually an ancient problem recast in modern terms. Hilbert asked if two polytopes P and Q can be each cut into a finite number of pieces P1, . . . , Pm and Q1, . . . , Qm with each Pi congruent to Qi by a rigid Euclidean motion, provided that P and Q have the same volume [18]. This question was answered in the negative by Max Dehn [2, 4, 29], who found a functional on polytopes that is invariant under dissections over rigid motions, while varying in value among polytopes of equal volume. In other words, the Dehn invariant is a “simple rigid motion invariant valuation” on polytopes that is not equal to volume (under any normalization). Dehn’s solution left open the question of exactly what conditions on P and Q imply equidissectability over the group of rigid motions, although this problem was solved by Hadwiger in the case where only translations (and no rotations nor reflections) are permitted (see [2, 16, 17, 26, 29]). In the course of studying this and related problems, Hadwiger discovered a characterization of Euclidean volume as a continuous rigid motion invariant simple valuation on compact convex sets, that is, a continuous rigid motion invariant valuation that vanishes on convex sets of less than full dimension. This result led in turn to a complete characterization of all continuous rigid motion invariant valuations on compact convex sets in R as consisting of a real (n + 1)-dimensional vector space spanned by the intrinsic volumes (or Quermassintegrals) [16] (also [20, 21, 31]). Since many standard functionals and integral operators can be interpreted as invariant valuations (such as intrinsic volumes, mean projections, Crofton and kinematic formulas), what came to be known as Hadwiger’s characterization theorem proved to be a valuable tool for generating quick and effortless proofs of many formulas and equations in integral geometry. Unfortunately Hadwiger’s original proof was long and difficult [16]. While seeking a shorter proof of Hadwiger’s volume characterization, the author discovered Received by the editors June 24, 1996 and, in revised form, September 29, 1997. 1991 Mathematics Subject Classification. Primary 52A22, 52A38, 52A39, 52B45. Research supported in part by NSF grants #DMS 9022140 to MSRI and #DMS 9626688 to the author. c ©1999 American Mathematical Society 71