2016/02/20 by Kyle Kinneberg, Kinneberg, Kyle
Mathematics · #FOS: Mathematics #Metric Geometry (math.MG) #Primary: 28A75 #Secondary: 30L99 #math.MG #msc:28A75 #msc:30L99
paper · pdf · doi:10.48550/arxiv.1602.06440
14 pages. v2: condensed and re-organized introduction, per referee comments; statements of main theorems changed accordingly, as well as Corollary 2.2; typos corrected. To appear in Rev. Mat. Iberoamericana
arxiv created 2016/10/21 · arxiv updated 2016/10/24
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 ≤ diam(M) have n-dimensional Hausdorff measure at least c ⋅ rn, where c>0 depends only on n and on the doubling and linear local contractibility constants.