  We study the robustness of single-field inflation against inhomogeneities. We
derive a simple analytic criterion on the shape of the potential for successful
inflation in the presence of inhomogeneities, and demonstrate its validity
using full 3+1 dimensional numerical relativity simulations on several classes
of popular models of single-field inflation. We find that models with convex
potentials are more robust to inhomogeneities than those with concave
potentials, and that concave potentials that vary on super-Planckian scales are
significantly more robust than those that vary on sub-Planckian scales.
