If $L(t,x,{\partial }_t,{\partial }_x)$ is a linear hyperbolic system of partial differential operators for which local uniqueness in the Cauchy problem at spacelike hypersurfaces is known, we find nearly optimal domains of determinacy of open sets ${\Omega }_0\subset \{t=0\}$. The frozen constant coefficient operators $L(\underline{t},\underline{x},{\partial }_t,{\partial }_x)$ determine local convex propagation cones, ${\Gamma }^{+}(\underline{t},\underline{x})$. Influence curves are curves whose tangent always lies in these cones. We prove that the set of points $\Omega$ which cannot be reached by influence curves beginning in the exterior of ${\Omega }_0$ is a domain of...