A $C^1$ regularity result for the inhomogeneous normalized infinity Laplacian

We prove that the unique solution to the Dirichlet problem with constant source term for the inhomogeneous normalized infinity Laplacian on a convex domain of ${\mathbb{R}}^N$ is of class $C^1$. The result is obtained by showing as an intermediate step the power-concavity (of exponent $1\mathrm{/}2$) of the solution.