Pogorelov type interior $C^2$ estimate for Hessian quotient equation and its application

In this paper, we derive a Pogorelov type interior $C^2$ estimate for the Hessian quotient equation $\frac{{\sigma }_n}{{\sigma }_k}\left(D^{2}u\right)=f$. As an application, we show that convex viscosity solutions are regular for $k\le n-3$ if $u\in C^{1,\alpha }$ with $\alpha >1-\frac{2}{n-k}$ or $u\in W^{2,p}$ with $p\ge \frac{(n-1)(n-k)}{2}$. Both exponents are sharp in view of the example in arXiv:2401.12229.