Regularity for convex viscosity solutions of ${\sigma }_2$ Equation

We prove interior $C^2$ regularity result for convex viscosity solutions of the quadratic Hessian equation ${\sigma }_2(D^{2}u)=f(x)$, under the assumption that $f\in C^{0,1}$ with $\inf f>0$. The result is almost sharp: if $f$ are merely continuous, there exist convex viscosity solutions that fail to be $C^{1,1}$. When $f\in C^{\alpha }$ for some $\alpha \in (0,1)$, the corresponding interior regularity remains open.