Continuous and profinite combinatorics

Gian-Carlo Rota believed that mathematics is a unity, in the deep sense that the same themes recur – as analogies – in its many branches. Thus, it comes as no surprise that Rota perceived combinatorial themes in continuous mathematics (as well as the other way around). In the preface to his book Introduction to Geometric Probability [10], written with the first author, Rota suggested only partly in jest that the field of geometric probability be renamed “continuous combinatorics”. Two of the papers reprinted in this chapter paper belong to the research program described in [10]. Perhaps the central idea behind Rota’s “continuous combinatorics” is the analogy between counting and measure, especially measures that are invariant with respect to some symmetry or group action. Here, the word “measure” is used in the broadest sense to include finitely additive measures which may admit no countably additive extension. These finitely additive measures, also called valuations, provide the intermediate hues in a spectrum of set functionals that extends from the purely discrete (such as lattice point enumerators and the Euler characteristic) to the analytic measures of Lebesgue theory. Although far more attention has been given in the last century to the two extreme cases (combinatorics and real analysis), constructions in convex and integral geometry going back as far as Minkowski offer a panoply of invariant valuations that are neither wholly analytic nor combinatorial in nature. Functionals on polytopes and convex sets such as the mean width, projection functions onto flats, and more general families of intrinsic volumes [10, 12], mixed volumes [18], and dual mixed volumes [11], provide examples whose fundamental properties are still poorly understood as compared to Lebesgue measure and simple counting. It was Rota’s contention that the best way to develop a comprehensive theory of these intermediate functionals is to determine how they connect analogous structures observed in combinatorics and real analysis, structures that are most evident in the contexts of combinatorial and analytic convex geometry. The analogies between the intrinsic volumes or Quermassintegrals (characterized by Hadwiger [6] as the fundamental valuations invariant under rigid motions), the Ehrhart coefficients of lattice polytopes (which are affine unimodular invariants later characterized by Betke and Kneser [1]), and fundamental families of enumerative functionals on simplicial complexes and finite vector spaces (for example, face and subspace enumerators [5, 8, 9, 10]), provide further evidence that a comprehensive theory of invariant set functionals is waiting in the wings. The motivation for the paper “Totally invariant set functions of polynomial type”, written with Beifang Chen, can be found in Problem Five in [14]. This problem, “Set functions on convex bodies”, is to prove the “correct” statement of the conjecture: