Differentiability of the argmin function and a minimum principle for semiconcave subsolutions

TL;DR

This paper proves the almost everywhere differentiability of the argmin function under certain convexity conditions and applies this to establish a minimum principle for specific semiconcave subsolutions.

Contribution

It introduces a differentiability result for the argmin function in a convex setting and derives a new minimum principle for semiconcave subsolutions.

Findings

Argmin function is differentiable almost everywhere.

Established a minimum principle for semiconcave subsolutions.

Provides theoretical foundation for optimization and analysis of subsolutions.

Abstract

Suppose $f(x,y) + \frac{\kappa}{2} \|x\|^2 - \frac{\sigma}{2}\|y\|^2$ is convex where $\sigma>0$, 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.