Revista Matemática Iberoamericana

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Volume 34, Issue 3, 2018, pp. 1103–1118
DOI: 10.4171/RMI/1018

Published online: 2018-08-27

Lower bounds for codimension-1 measure in metric manifolds

Kyle Kinneberg[1]

(1) National Security Agency, Fort Meade, USA

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.

Keywords: Hausdorff measure, isoperimetric inequality, linearly locally contractible metric manifold

Kinneberg Kyle: Lower bounds for codimension-1 measure in metric manifolds. Rev. Mat. Iberoam. 34 (2018), 1103-1118. doi: 10.4171/RMI/1018