Discrete Curvatures and Convex Polytopes

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 $\mathrm{\Delta }$-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.