Defining Geodesic Convexity
In my previous blog post , I discussed the notion of geodesic convexity , which is a generalization of ordinary convexity in Euclidean space to Riemannian manifolds. This is essential for disciplined optimization over manifolds; hence, it's very relevant to my research in optimization-based motion planning over manifolds. Unfortunately, in the current literature, there are various definitions of geodesic convexity, with key differing properties. This is chiefly because if a geodesic connecting two points exists (it may not exist), it may not be unique, and it may not be minimizing. In this brief blog post, I'll discuss the various definitions that are used, and then explain why we use the one we did in our GGCS paper . The Definitions Nicholas Boumal's textbook An introduction to optimization on smooth manifolds is an excellent reference on the subject, and he discusses the various definitions of geodesic convexity in Section 11.3. The first definition of geodesic convexi