Differentiability of the Argmin Function and a Minimum Principle for Semiconcave Subsolutions

Suppose $f(x,y) + \frac{\kappa}{2} \|x\|^2 - \frac{\sigma}{2}\|y\|^2$ is convex where $\kappa\ge 0, \sigma>0$, and the argmin function $\gamma(x) = \{ \gamma: \inf_y f(x,y) = f(x,\gamma)\}$ exists and is single valued. We will prove $\gamma$ is differentiable almost everywhere. As an application we deduce a minimum principle for certain semiconcave subsolutions.