2019/05/16 by Yevgeny Liokumovich, Boris Lishak, Liokumovich, Yevgeny +5 · 1 citation
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG) #math.DG #math.MG
paper · pdf · doi:10.48550/arxiv.1905.06522
arxiv created 2021/02/24 · arxiv updated 2021/02/26
We prove a new version of isoperimetric inequality: Given a positive real m, a Banach space B, a closed subset Y of metric space X and a continuous map f:Y → B with f(Y) compact infFHCm+1(F(X))≤ c(m)HCm(f(Y))(m+1)/(m), where HCm denotes the m-dimensional Hausdorff content, the infimum is taken over the set of all continuous maps F:X\longrightarrow B such that F(y)=f(y) for all y∈ Y, and c(m) depends only on m. Moreover, one can find F with a nearly minimal HCm+1 such that its image lies in the C(m)HCm(f(Y))1\over m-neighbourhood of f(Y) with the exception of a subset with zero (m+1)-dimensional Hausdorff measure. The paper also contains a very general coarea inequality for Hausdorff content and its modifications. As an application we demonstrate an inequality conjectured by Larry Guth that relates the m-dimensional Hausdorff content of a compact metric space with its (m-1)-dimensional Urysohn width. We show that this result implies new systolic inequalities that both strengthen the classical Gromov's systolic inequality for essential Riemannian manifolds and extend this inequality to a wider class of non-simply connected manifolds.