Semiconcave Functions in Alexandrov’s Geometry

Abstract

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

Geometric and Topological Inference

“…Distance functions are also particular cases of the so-called semiconcave functions. Many of the results presented in this section can be deduced from general properties of semiconcave functions [117]. This allows in particular to extend most of the results given in this section to compact subsets of Riemannian manifolds.…”

“…The notion of critical point introduced in this chapter coincides with the notion of critical point for distance function used in riemannian geometry and non-smooth analysis where the notion of Clarke gradient [57] is closely related to the above defined gradient. The general properties of the gradient of d K and of its trajectories given in Section 9.2, are established in [101] or, in a more general setting, in [117]. Distance functions are also particular cases of the so-called semiconcave functions.…”

“…In other words, ∇d K is a discontinuous vector field. Nevertheless it is possible to show [101,117] that x → ∇d K (x) is a lower semi-continuous function, i.e. for any a ∈ R, ∇d K −1 ((−∞, a]) is a closed subset of R d .…”

Discrete-time gradient flows and law of large numbers in Alexandrov spaces

“…We cannot expect these properties in the lower curvature bound case, the injectivity radius can be 0 even locally. Then it is difficult to control the behavior of discrete-time flows and there are no investigation in this direction as far as the authors know, while continuous-time gradient flows are intensively studied in [37,38,29,35,19].…”

“…An Alexandrov space is a metric space whose sectional curvature is bounded above or below by some constant in the sense of triangle comparison theorem (see Section 2). The where g-exp is the gradient exponential map and ∇(−f )(x) denotes the gradient vector of −f (see [37,38] and Sections 3, 4 for these notions). Before discussing the reason why we use these different resolvents, we present our results in this paper.…”