Generalized asymptotic Euler's relation for certain families of polytopes

TL;DR

This paper generalizes Euler's relation for polytopes by exploring asymptotic face count proportions based on modular dimension classes, revealing that certain polytopes exhibit a uniform distribution of face dimensions in the limit.

Contribution

It introduces a generalized asymptotic relation for face counts of polytopes based on modular dimension classes, extending classical Euler's relation.

Findings

For certain classes of polytopes, the proportion of faces with dimensions congruent to i mod m approaches 1/m.

The paper establishes asymptotic equal distribution of face dimensions among classes for these polytopes.

Provides new insights into the combinatorial structure of polytopes beyond classical Euler's relation.

Abstract

According to Euler's relation any polytope P has as many faces of even dimension as it has faces of odd dimension. As a generalization of this fact one can compare the number of faces whose dimension is congruent to i modulo m with the number of all faces of P for some positive integer m and for some 0 < i < m+1. We show some classes of polytopes for which the above proportion is asymptotically equal to 1/m.