Geodesic
Differentiability of the Argmin Function and a Minimum Principle for Semiconcave Subsolutions
Journal of convex analysis, Tome 27 (2020) no. 3, pp. 811-832
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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.
Classification : 28B20, 58C06
Mots-clés : Argmin function, differentiability, minimum principle, semiconcave subsolutions 
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     author = {J. Ross and D. Witt Nystr\"om},
     title = {Differentiability of the {Argmin} {Function} and a {Minimum} {Principle} for {Semiconcave} {Subsolutions}},
     journal = {Journal of convex analysis},
     pages = {811--832},
     year = {2020},
     volume = {27},
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J. Ross; D. Witt Nyström. Differentiability of the Argmin Function and a Minimum Principle for Semiconcave Subsolutions. Journal of convex analysis, Tome 27 (2020) no. 3, pp. 811-832. http://geodesic.mathdoc.fr/item/JCA_2020_27_3_JCA_2020_27_3_a1/