In this paper, we prove a theorem on tight paths in convex geometric hypergraphs, which is asymptotically sharp in infinitely many cases. Our geometric theorem is a common generalization of early results of Hopf and Pannwitz [12], Sutherland [19], Kupitz and Perles [16] for convex geometric graphs, as well as the classical Erdős-Gallai Theorem [6] for graphs. As a consequence, we obtain the first substantial improvement on the Turán problem for tight paths in uniform hypergraphs.