On the exponential convergence of Kobayashi geodesics in strongly convex domains

In this paper, we have proved a qualitative version of the approaching geodesic property for certain convex domains. More precisely, we have proved that that if $\Omega \subset \mathbb{C}^{d}$ is a bounded strongly convex domain with $\mathcal{C}^3$ boundary and $\gamma_{1}, \gamma_{2}:[0, \infty) \to \Omega$ are two geodesics such that $\gamma_{1}(\infty)=\gamma_{2}(\infty)=\xi \in \partial \Omega$. Then there exists $A\big(\gamma_{1}(0), \gamma_{2}(0) \big)>0$ and $T \in \mathbb{R}$ such that
$$K_{\Omega}(\gamma_{1}(t), \gamma_{2}(t+T))\leq Ae^{-\frac{t}{2}} ~~~\,\hspace{1em} \forall t \geq 0.$$ Furthermore, using this property we provided a characterization of strongly pseudoconvex domain via a biholomorphic invariant function namely generalized squeezing function. We have proved that: For every $\alpha>2$ there exists $\epsilon(d,\alpha)>0$ such that the following holds: if $\Omega \subset \mathbb{C}^d$ is a bounded convex domain with $\mathcal{C}^{2,\alpha}$-boundary and \[ T_{\Omega}^{D}(z)\geq 1-\epsilon \] outside a compact subset of $\Omega$, where $D \Subset \mathbb{C}^{d}$ is a balanced strongly convex domain with $\mathcal{C}^{3}$ boundary and $T_{\Omega}^{D}$ is the squeezing function of $\Omega$ with respect to the domain $D$ then $\Omega$ is strongly pseudoconvex.