Uniform Interior $C^{1,\alpha }$ Estimates for Infinity-Harmonic Functions

We prove a uniform interior $C^{1,{\alpha }_d}$ estimate for bounded infinity-harmonic functions in every dimension d ≥ 3. For some ${\alpha }_d\in (0,1/3]$ depending only on the dimension, the gradient's supremum norm and α${}_d$-Hölder seminorm on B${}_{1/2}$ are bounded by a dimension-dependent constant times the oscillation on B${}_1$. The exponent is not explicit, and no endpoint or boundary regularity claim is made. In particular, infinity-harmonic functions are locally C${}^1$ on every open subset of ℝ${}^d$, for d ≥ 3.