It is shown that a greedy orthogonal access scheme achieves the sum degrees of freedom of all one-dimensional (all nodes placed along a straight line) convex cellular networks (where cells are convex regions) when no channel knowledge is available at the transmitters except the knowledge of the network topology. In general, optimality of orthogonal access holds neither for two-dimensional convex cellular networks nor for one-dimensional non-convex cellular networks, thus revealing a fundamental limitation that exists only when both one-dimensional and convex properties are simultaneously enforced, as is common in canonical information theoretic models for studying cellular networks. The result also establishes the capacity of the corresponding class of index coding problems.