Hilbert Volume in Metric Spaces

Abstract

We introduce a notion of Hilbertian n-volume in metric spaces with Besicovitch-type inequalities built-in into the definitions. The present Part 1 of the article is, for the most part, dedicated to reformulation of known results in our terms with proofs being reduced to (almost) pure tautologies. If there is any novelty in the paper, this is in forging certain terminology which, ultimately, may turn useful in Alexandrov kind of approach to singular spaces with positive scalar curvature [17].

Product Inequalities for $\mathbb T^\rtimes$-Stabilized Scalar Curvature

“…and Sc Moreover, the fundamental classes of compact connected manifolds X with non-empty boundaries are strictly positive since such manifolds admit metrics with Sc > 0. (For instance, the r-balls with hyperbolic metrics g, sect.curv(g) = −1, admit (obvious radial) metrics g + ≥ g with Sc(g 15 2.H. Finiteness.…”

“…At the same time, much of the Dirac theoretic scalar curvature results apply to these X, see [13, Section 91 2 ] 9. This makes sense for general metric spaces X with the "Hilbertian area" defined in[15].…”

“…It is unclear what happens form ≤ n ≤ 2m − 2.13 Y ⊂ X is homogeneous if an isometry group of X preserves Y and is transitive on Y 14. The dimension m = 4 may be special 15. This seems more realistic if β can be homotoped to the (m − 2)-skeleton of (some cell decomposition of) B(π1(X)).…”