A partial result: if a planar curve intersects every line in at
most 3 points, then it can be partitioned into 4 convex curves. This result can be extended to R^d: if a curve in R^d intersects every hyperplane at most d+1 times, then it can be split into M(d) convex curves. The extension implies a good, asymptotically precise, lower bound on a geometric Ramsey number.
Joint result with Jiri Matousek and Attila Por.