Good coverings of proximal Alexandrov spaces. Path cycles in the extension of the Mitsuishi-Yamaguchi good covering and Jordan Curve Theorems by James Francis Peters, Tane Vergili

“…This paper introduces proximal path cycles, which lead to the main results in this paper, namely, extensions of the Mitsuishi-Yamaguchi Good Coverning Theorem with different forms of Tanaka good cover of an Alexandrov space equipped with a proximity relation as well as extension of the Jordan curve theorem. …”
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Extremal subsets in geodesically complete spaces with curvature bounded above by Fujioka Tadashi

“…This is an analog of an extremal subset in an Alexandrov space with curvature bounded below introduced by Perelman and Petrunin. …”
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A note on flatness of non separable tangent cone at a barycenter by Le Gouic, Thibaut

“…Given a probability measure $\mathbf{P}$ on an Alexandrov space $S$ with curvature bounded below, we prove that the support of the pushforward of $\mathbf{P}$ on the tangent cone $T_{b^\star }S$ at its (exponential) barycenter $b^\star $ is a subset of a Hilbert space, without separability of the tangent cone.…”
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