Fast convergence of empirical barycenters in Alexandrov spaces and the Wasserstein space by Le Gouic, Thibaut, Paris, Quentin, Rigollet, Philippe, Stromme, Austin J

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Sharp spectral gap and Li-Yau's estimate on Alexandrov spaces by Qian, Z, Zhang, H, Zhu, X

Sharp spectral gap and Li-Yau's estimate on Alexandrov spaces by Qian, Z, Zhang, H, Zhu, X

“…Firstly, we extend the sharp lower bound estimates of spectral gap, due to Chen and Wang (Sci Sin (A) 37:1-14, 1994), Chen and Wang (Sci Sin (A) 40:384-394, 1997) and Bakry-Qian (Adv Math 155:98-153, 2000), from smooth Riemannian manifolds to Alexandrov spaces. As an application, we get an Obata type theorem for Alexandrov spaces. …”

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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Measure rigidity of Ricci curvature lower bounds by Cavalletti, F, Mondino, A

“…The main example will be the Hausdorff measure over an Alexandrov space. Then we prove that the following are reference measures over smooth spaces: the volume measure of a Riemannian manifold, the Hausdorff measure of an Alexandrov space with bounded curvature, and the Haar measure of the subRiemannian Heisenberg group.…”

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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On quotients of spaces with Ricci curvature bounded below by Galaz-García, F, Kell, M, Mondino, A, Sosa, G

“…It is well known that a lower bound of the sectional curvature of (M,g) is again a bound for the curvature of the quotient space, which is an Alexandrov space of curvature bounded below. Moreover, the analogous stability property holds for metric foliations and submersions. …”

The metric measure boundary of spaces with Ricci curvature bounded below by Brué, E, Mondino, A, Semola, D

“…Our result, combined with [KLP21], settles an open question about the existence of infinite geodesics on Alexandrov spaces without boundary raised by Perelman and Petrunin in 1996.…”

On the volume measure of non-smooth spaces with Ricci curvature bounded below by Kell, M, Mondino, A

“…We also show that a Lipschitz differentiability space which is bi-Lipschitz embeddable into a euclidean space is rectifiable as a metric measure space, and we conclude with an application to Alexandrov spaces.…”

Angles between curves in metric measure spaces by Han, B, Mondino, A

“…As a consequence, the new introduced notions are compatible with the corresponding classical ones for Riemannian manifolds, Ricci limit spaces and Alexandrov spaces.…”

Polya-Szego inequality and Dirichlet p-spectral gap for non-smooth spaces with Ricci curvature bounded below by Mondino, A, Semola, D

“…This extends to the non-smooth setting a classical result of Bérard-Meyer [14] and Matei [41]; remarkable examples of spaces fitting our framework and for which the results seem new include: measured-Gromov Hausdorff limits of Riemannian manifolds with Ricci , finite dimensional Alexandrov spaces with curvature, Finsler manifolds with Ricci . …”

Weak Laplacian bounds and minimal boundaries in non-smooth spaces with Ricci curvature lower bounds by Mondino, A, Semola, D

“…The class of RCD(K,N) metric measure spaces includes as remarkable sub-classes: measured Gromov-Hausdorff limits of smooth manifolds with lower Ricci curvature bounds and finite dimensional Alexandrov spaces with lower sectional curvature bounds. …”

Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds by Cavalletti, F, Mondino, A

“…Examples of spaces fitting our assumptions include measured Gromov–Hausdorff limits of Riemannian manifolds satisfying Ricci curvature lower bounds, Alexandrov spaces with curvature bounded from below, Finsler manifolds endowed with a strongly convex norm and satisfying Ricci curvature lower bounds; the result seems new even in these celebrated classes of spaces.…”

Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds by Cavalletti, F, Mondino, A

“…Examples of spaces entering this framework are (weighted) Riemannian manifolds satisfying lower Ricci curvature bounds and their measured Gromov Hausdorff limits, Alexandrov spaces satisfying lower curvature bounds and, more generally, RCD*(K,N) spaces, Finsler manifolds endowed with a strongly convex norm and satisfying lower Ricci curvature bounds. …”