Discrete Curvatures and Convex Polytopes

Abstract.

We study Forman–Ricci and effective resistance curvatures on the skeleta of convex polytopes. Our guiding questions are, How frequently do polytopal graphs exhibit everywhere-positive curvature, and what structural constraints does positivity impose? For Forman–Ricci curvature, we derive an exact identity for the average edge curvature in terms of flag [Formula: see text]-numbers and establish the existence of infinite families of Forman–Ricci-positive polytopes in every fixed dimension [Formula: see text]. We prove finiteness results in low dimension: there are only finitely many Forman–Ricci-positive 3- and 4-polytopes; for [Formula: see text] we show finiteness in the simplicial case and conjecture its extension to 5-polytopes more generally. For the resistance curvature [Formula: see text] we establish the existence of infinite families of resistance-positive polyhedral graphs for all [Formula: see text], and we provide a quantitative lower bound for [Formula: see text] in a simple 3-polytope in terms of the lengths of the three 2-faces incident to [Formula: see text]. This bound leads to constructions of non-vertex transitive, resistance-positive 3-polytopes via [Formula: see text]-operations, and a degree-based obstruction showing that if each neighbor of [Formula: see text] has degree at most [Formula: see text], then [Formula: see text]. Our results suggest that positive curvature on polytopal skeleta is rare and constrained.