The average simplex cardinality of a finite abstract simplicial complex

Oliver Knill

Published 2019-05-06Version 1

We study the average simplex cardinality Dim^+(G) = sum_x |x|/(|G|+1) of a finite abstract simplicial complex G. The functional is a homomorphism from the monoid of simplicial complexes to the rationals: the formula Dim^+(G + H) = Dim^+(G) + Dim^+(H) holds for the join + similarly as for the augmented inductive dimension dim^+(G) = dim(G)+1 where dim is the inductive dimension dim(G) = 1+ sum_x dim(S(x))/|G| with unit sphere S(x) (a recent theorem of Betre and Salinger). In terms of the generating function f(t) = 1+v_0 t + v_1 t^2 + ... +v_d t^(d+1) defined by the f-vector (v_0,v_1, \dots) of G for which f(-1) is the genus 1-X(G) with Euler characteristic X and f(1)=|G|+1 is the augmented number of simplices, the average cardinality is the logarithmic derivative Dim^+(f) = f'(1)/f(1) of f at 1. Beside introducing the average cardinality and establishing its compatibility with arithmetic, we prove two results: 1) the inequality dim^+(G)/2 <= Dim^+(G) with equality for complete complexes. 2) the limit C_d of Dim^+(G_n) for n to infinity is the same for any initial complex G_0 of maximal dimension d and the constant c_d is explicitly given in terms of the Perron-Frobenius eigenfunction of the universal Barycentric refinement operator and is for positive d always a rational number in the open interval ((d+1)/2,d+1).