  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, Sutherland, Kupitz and Perles for convex geometric graphs, as well as
the classical Erd\H{o}s-Gallai Theorem for graphs. As a consequence, we obtain
the first substantial improvement on the Tur\'{a}n problem for tight paths in
uniform hypergraphs.
