Geodesic bicombing

In metric geometry, a geodesic bicombing distinguishes a class of geodesics of a metric space. The study of metric spaces with distinguished geodesics traces back to the work of the mathematician Herbert Busemann. The convention to call a collection of paths of a metric space bicombing is due to William Thurston. By imposing a weak global non-positive curvature condition on a geodesic bicombing several results from the theory of CAT(0) spaces and Banach space theory may be recovered in a more general setting.

Definition

Let be a metric space. A map is a geodesic bicombing if for all points the map is a unit speed metric geodesic from to , that is, , and for all real numbers .

Different classes of geodesic bicombings

A geodesic bicombing is:

• reversible if for all and .
• consistent if whenever and .

• conical if for all and .
• convex if is a convex function on for all .

Examples

Examples of metric spaces with a conical geodesic bicombing include:

• Banach spaces.
• CAT(0) spaces.
• injective metric spaces.
• the spaces where is the first Wasserstein distance.
• any ultralimit or 1-Lipschitz retraction of the above.

Properties

• Every consistent conical geodesic bicombing is convex.
• Every convex geodesic bicombing is conical, but the reverse implication does not hold in general.
• Every proper metric space with a conical geodesic bicombing admits a convex geodesic bicombing.
• Every complete metric space with a conical geodesic bicombing admits a reversible conical geodesic bicombing.

References