Lower bounds for codimension-1 measure in metric manifolds

Abstract

We establish Euclidean-type lower bounds for the codimension-1 Hausdorff measure of sets that separate points in doubling and linearly locally contractible metric manifolds. This gives a quantitative topological isoperimetric inequality in the setting of metric manifolds, in the sense that lower bounds for the codimension-1 measure of a set depend not on some notion of filling or volume but rather on in-radii of complementary components. As a consequence, we show that balls in a closed, connected, doubling, and linearly locally contractible metric $n$-manifold $(M,d)$ with radius $0<r \leq \text{diam}(M)$ have $n$-dimensional Hausdorff measure at least $c \cdot r^n$, where $c>0$ depends only on $n$ and on the doubling and linear local contractibility constants.