At a point p of a Riemannian manifold, the convexity radius is the largest r>0 such that the geodesic ball B(p,r) is (strongly) geodesically convex: any two points in B(p,r) are joined by a unique minimizing geodesic lying entirely in B(p,r).

Principle

Strengthens injectivity by requiring pairwise uniqueness of minimizing geodesics inside the ball rather than merely the exponential map's local diffeomorphism; it controls the ball's geodesic convexity and variational uniqueness properties.

Demonstration

On the standard sphere of radius R the convexity radius at a point is πR/2 because beyond that distance two points in the ball can be joined by distinct minimizing geodesics via antipodal regions; on manifolds with curvature bounds one can give positive lower estimates from comparison theorems.

Misapplication

Assuming a small metric ball in coordinates is geodesically convex without checking curvature and cut-locus phenomena, or conflating convexity radius with injectivity radius though they differ in general.

Consequence

Inside the convexity radius, distance-minimizing geodesics between any pair of points in the ball are unique and depend smoothly on endpoints, enabling convex analysis, unique projection properties and simplifications of variational problems.

Reversal

Outside this radius geodesic convexity can fail: there may be multiple minimizers between two nearby points, and geodesic balls can exhibit nonconvex shape due to focal or cut points.

Boundary

Depends on the base point and the global curvature/topology; convexity radius is always bounded above by the injectivity radius and by quantities determined by conjugate points and topology—it is not an extrinsic notion and applies to Riemannian metrics only.

Semantic Tension

Closely related to but distinct from injectivity radius and from extrinsic notions like reach; some authors use weaker convexity definitions (local versus strong geodesic convexity) so terminology must be checked in context.

Synthesis

The convexity radius quantifies the largest geodesic ball around a point in which geodesic minimizers between any two points are unique and contained in the ball, providing a controlled domain for convex geometric and variational arguments.