21 research outputs found
Characterization of Low Dimensional spaces
In this paper, we give the characterization of metric measure spaces that
satisfy synthetic lower Riemannian Ricci curvature bounds (so called
spaces) with \emph{non-empty} one dimensional regular sets. In
particular, we prove that the class of Ricci limit spaces with and
Hausdorff dimension and the class of spaces coincide for (They can be either complete intervals or circles). We will also prove a
Bishop-Gromov type inequality ( that is ,roughly speaking, a converse to the
L\'{e}vy-Gromov's isoperimetric inequality and was previously only known for
Ricci limit spaces) which might be also of independent interest.Comment: version 3: 37 pp, to appear in AGM