On local quadratic subconvexity of metrics and rectifiability of Bezier curves in CAT(K) spaces

Abstract

We generalize the definition of Bezier curves (which agrees with the usual one in the Euclidean case) in CAT($K$) spaces. Rectifiability---a naturally-expected property of B\'ezier curves---is proved in this general setting. In order to achieve this, an assertion weaker but similar to convexity of metrics of CAT($0$) spaces is verified for general metric spaces of curvature bounded above.