The scalar curvature of a projectively invariant metric defined by the kernel function

Considering a projectively invariant metric $\tau$ defined by the kernel function on strongly convex bounded domain $\Omega\subset\mathbb{R}^n$, we study asymptotic expansion of scalar curvature with respect to distance function, and use Fubini-Pick describe second term in expansion. This implies that if $n\geq 3$ $(\Omega,\tau )$ has constant curvature, then is equivalent ball.