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

A proof of the Krylov-Safonov theorem without localization