Semiconcave Functions in Alexandrov???s Geometry

By Anton Petrunin. 2013

198 citations recorded by the source corpus.

The following is a compilation of some techniques in Alexandrov's geometry which are directly connected to convexity.

Subfields: Applied Mathematics, Computational Mechanics, Computer Vision and Pattern Recognition

View Paper: https://www.semanticscholar.org/paper/d1a463931e673e7ef007aac9de6e8678a0bf1915

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In Computational Mechanics

In Computer Vision and Pattern Recognition

Citations in the corpus

An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists (2017, background): “Thanks to its semiconcavity, a distance-like function φ have a well-defined, but not continuous, gradient ∇φ : Rd → Rd that can be integrated into a continuous flow (Petrunin, 2007) that allows to track the evolution of the topology of its sublevel sets and to compare it to the one of the sublevel s”
Displacement convexity of generalized relative entropies (2010, methodology): “In this section, we adapt the rather ‘metric geometric’ approach developed in [Oh1] inspired by [PP] and [Ly] (see also [Pe]).”
Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows (2018, background): “ theory of Alexandrov spaces, where lower bounds on the sectional curvature turn into the existence of many semi-convex functions thus having, at least formally, bounded second order derivatives (see [P07]). Nevertheless, having at disposal a rich family of vector ﬁelds, one can start investigating ”
Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics (2018, methodology): “Following Petrunin (2007), we denote ⇑xp⊂ Σp the set of all directions (or classes) of constant speed shortest paths from p to x and, for all x ∈ S, we select ↑xp∈⇑xp one such direction.”
Gradient flows and Evolution Variational Inequalities in metric spaces. I: Structural properties (2018, unknown): “PC spaces, or more generally Alexandrov spaces with a uniform lower curvature bound [23, 22, 79, 77, 78, 82, 72, 70], .”
Non-collapsed spaces with Ricci curvature bounded from below (2017, unknown): “(notice that the analogous of this latter question for Alexandrov spaces is open - see [48, Section 9]).”
Perelman???s Stability Theorem (2007, methodology): “Alternatively, it was shown in [Pet06] that E is extremal iff it’s invariant under gradient flows of all semiconcave functions on X .”
Geodesically complete spaces with an upper curvature bound (2018, background): “tz continuous convex functions. Due to the corresponding statements about DC-functions on intervals, the set of DC functions on Y is closed under addition and multiplication. Remark 14.2. We refer to [Pet07] for the deﬁnition and properties of semiconvexity. Assume that in Y each point x admits a Li”
Ricci Curvature on Alexandrov spaces and Rigidity Theorems (2009, methodology): “We refer to Sections 1 and 2 in [32] for the details on semi-concave functions, gradient curves and gradient flows.”
A weighted k-nearest neighbor density estimate for geometric inference (2011, background): “:3) x 2 − d 2 μ,m 0 ( x ) is convex, see Petrunin [40] for more information on geometric properties of semiconcave functions).”
Nilpotency, almost nonnegative curvature, and the gradient flow on Alexandrov spaces (2005, background): “his article. 2. Alexandrov geometry and the gradient flow This section provides necessary background in Alexandrov geometry. The results ofsections 2.1–2.3are mostly repeated from [PP96], [Pet95] and [Pet07]. The reader may consult [BGP92] for a general reference on Alexandrov spaces. NILPOTENCY, AL”
A knot characterization and 1-connected nonnegatively curved 4-manifolds with circle symmetry (2013, background): “Such sets play an important role in Alexandrov geometry; see [17] for further information.”