  The goal of this paper is to continue the study of the relation between the
Poincar\'e inequality and the lower bounds of Minkowski content of separating
sets, initiated in our previous work [Caputo, Cavallucci: Poincar\'e inequality
and energy of separating sets, arXiv 2401.02762]. A new shorter proof is
provided. An intermediate tool is the study of the lower bound of another
geometric quantity, called separating ratio. The main novelty is the
description of the relation between the infima of the separating ratio and the
Minkowski content of separating sets. We prove a quantitative comparison
between the two infima in the local quasigeodesic case and equality in the
local geodesic one. No Poincar\'e assumption is needed to prove it. The main
tool employed in the proof is a new function, called the position function,
which allows in a certain sense to fibrate a set in boundaries of separating
sets. We also extend the proof to measure graphs, where due to the
combinatorial nature of the problem, the approach is more intuitive. In the
appendix, we revise some classical characterizations of the p-Poincar\'e
inequality, by proving along the way equivalence with a notion of p-pencil that
extends naturally the definition for p = 1.
