Alexandrov geometry: foundations

Asymptotic topological regularity of CAT(0) spaces

“…For each i, let U 1 (p; λ i X) be the open metric ball of radius 1 around p in λ i X. By Proposition 3.4 (2), for the open ball U 1 (0) in C 0 (∂ T X) we see dim U 1 (0) = n. Therefore from Theorem 2.8 we derive…”

“…We refer the readers to [1], [2], [3], [5], [8], [10], [11] for the basic facts on metric spaces with an upper curvature bound.…”

5-point Toponogov theorem

“…The (n+1)-comparison is another condition that holds for any (n + 1)-point array in Alexandrov spaces [1,2]. It says that given a point array p, x 1 , .…”

Topological representation of the geometry of metric spaces

“…If we have a data set X in the Euclidean space E d and U = {B(x α , ǫ), α ∈ I, x α ∈ F ⊂ X, ǫ > 0} is a cover of closed balls for X, then the nerve of U is topologically equivalent to the ǫ-neighborhood of F in X. 1 The concept of a nerve was introduced by Alexandroff [2], and nerve theorems for sufficiently good (e.g. paracompact) topological spaces were proved by Borsuk [8], Leray [23] and Weil [28].…”

“…Gromov [15] developed a general theory of hyperbolic spaces, that is, spaces whose curvature is negative. Abstract versions of curvature inequalities involve relations between the distances in certain configurations of 4 points, see for instance [1,7,5]. Another important theory, developed by Isbell [17] and Dress [12] and others, e.g.…”