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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 Teichmüller space
T
g
,
n
is not biholomorphic to any bounded pseudoconvex domain in
C
3
g
-
3
+
n
which is locally log-type convex near some boundary point.