  In this paper we introduce a new class of domains -- log-type convex domains,
which have no boundary regularity assumptions. Then we will localize the
Kobayashi metric in log-type convex subdomains. As an application, we prove a
local version of continuous extension of rough isometric maps between two
bounded domains with log-type convex Dini-smooth boundary points. Moreover we
prove that the Teichm\"uller space $\mathcal T_{g,n}$ is not biholomorphic to
any bounded pseudoconvex domain in $\mathbb C^{3g-3+n}$ which is locally
log-type convex near some boundary point.
