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 $f$-numbers and establish the existence of infinite families of Forman--Ricci-positive polytopes in every fixed dimension $d\ge 6$. We prove finiteness results in low dimension: there are only finitely many Forman--Ricci-positive $3$- and $4$-polytopes; for $d=5$ we show finiteness in the simplicial case, and conjecture its extension to $5$-polytopes more generally. For the resistance curvature $\kappa(v)$ we establish the existence of infinite families for all $d\ge 3$, and we provide a quantitative lower bound for $\kappa(v)$ in a simple $3$-polytope in terms of the lengths of the three $2$-faces incident to $v$. This bound leads to constructions of non-vertex-transitive, resistance-positive $3$-polytopes via $\Delta$-operations, and a degree-based obstruction showing that if each neighbor of $v$ has degree at most $d_v-2$, then $\kappa(v)\le 0$. Our results suggest that positive curvature on polytopal skeletons is rare and constrained.