CD meets CAT

We show that if a noncollapsed {\mathrm{CD}(K,n)} space X with {n\geq 2} has curvature bounded above by κ in the sense of Alexandrov, then {K\leq(n-1)\kappa} and X is an Alexandrov space of curvature bounded below by {{K}-\kappa(n-2)}. We also show that if a {\mathrm{CD}(K,n)} space Y with finite n has curvature bounded above, then it is infinitesimally Hilbertian.

Absolute Continuity of Wasserstein Barycenters Over Alexandrov Spaces

In this paper, we prove that on a compact, n-dimensional Alexandrov space with curvature at least −1, the Wasserstein barycenter of Borel probability measures μ1 ,… , μm is absolutely continuous with respect to the n-dimensional Hausdorff measure if one of them is.

Generalized packing radius theorems of Alexandrov spaces with curvature ≥ 1

In this paper, we give some generalized packing radius theorems of an [Formula: see text]-dimensional Alexandrov space [Formula: see text] with curvature [Formula: see text]. Let [Formula: see text] be any [Formula: see text]-separated subset in [Formula: see text] (i.e. the distance [Formula: see text] for any [Formula: see text]). Under the condition “[Formula: see text]” (after [K. Grove and F. Wilhelm, Hard and soft packing radius theorems, Ann. of Math. 142 (1995) 213–237]), we give the upper bound of [Formula: see text] (which depends only on [Formula: see text]), and classify the geometric structure of [Formula: see text] when [Formula: see text] attains the upper bound. As a corollary, we get an isometrical sphere theorem in Riemannian case.

A rigidity theorem for discrete groups

This work considers the discrete subgroups of group of isometries of an Alexandrov space with a lower curvature bound. By developing the notion of Hausdorff distance in these groups, a rigidity theorem for the close discrete groups was proved.

ALEXANDROV SPACES WITH SYSTOLS ALMOST TWICE THE DIAMETERS

Let X be an Alexandrov space with Systol(X)=2 and diameter at a point almost 1, we prove that the fundamental group of X must be Z2.