Lower bounds for codimension-1 measure in metric manifolds

Abstract

We establish Euclidean-type lower bounds for the codimen\-sion-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 \mathrm {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.