Lower bounds for codimension-1 measure in metric manifolds

TL;DR

This paper proves lower bounds for the measure of sets that separate points in certain metric manifolds, establishing a quantitative isoperimetric inequality based on in-radii, with implications for the measure of metric balls.

Contribution

It introduces Euclidean-type lower bounds for codimension-1 measures in metric manifolds, linking topological separation to geometric measure bounds.

Findings

Lower bounds depend on in-radii of complementary components.

Balls in the manifold have measure at least proportional to their radius to the power n.

Provides a quantitative isoperimetric inequality 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.