Lower bounds for codimension-1 measure in metric manifolds

  • Kyle Kinneberg

    National Security Agency, Fort Meade, USA
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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 nn-manifold (M,d)(M,d) with radius 0<rdiam(M)0 < r \leq \mathrm {diam}(M) have nn-dimensional Hausdorff measure at least~crnc \cdot r^n, where c>0c>0 depends only on nn and on the doubling and linear local contractibility constants.

Cite this article

Kyle Kinneberg, Lower bounds for codimension-1 measure in metric manifolds. Rev. Mat. Iberoam. 34 (2018), no. 3, pp. 1103–1118

DOI 10.4171/RMI/1018