So with a few minor complications convexity generalises to Riemannian manifolds like the earth. You need to replace "straight line" with "minimising geodesic" i.e. shortest path, which don't depend on the choice of coordinate chart, just on the Riemannian manifold structure (which includes an inner product hence a metric).

This is complicated slightly when there isn't a unique shortest path between any given two points (e.g. the earth's north and south poles), leading to definitions of strongly convex, convex and weakly convex. See http://en.wikipedia.org/wiki/Geodesic_convexity and the debate at http://en.wikipedia.org/wiki/Talk:Geodesic_convexity#Dispute...