FILLING METRIC SPACES

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 (R) B with f(Y) compact inf(F)HC(m+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 (R) B such that F(y) = f(y) for all y I 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/m)-neighborhood 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.