  The Krylov-Safonov theorem says that solutions to non-divergence uniformly
elliptic equations with rough coefficients are H\"{o}lder continuous. The proof
combines a basic measure estimate with delicate localization and covering
arguments. Here we give a "global" proof based on convex analysis that avoids
the localization and covering arguments. As an application of the technique we
prove a $W^{2,\,\epsilon}$ estimate where $\epsilon$ decays with the
ellipticity ratio of the coefficients at a rate that improves previous results,
and is optimal in two dimensions.
