  Teichm\"uller's classical mapping problem for plane domains concerns finding
a lower bound for the maximal dilatation of a quasiconformal homeomorphism
which holds the boundary pointwise fixed, maps the domain onto itself, and maps
a given point of the domain to another given point of the domain. For a domain
$D \subset {\mathbb R}^n\,,n\ge 2\,,$ we consider the class of all $K$-
quasiconformal maps of $D$ onto itself with identity boundary values and
Teichm\"uller's problem in this context. Given a map $f$ of this class and a
point $x\in D\,,$ we show that the maximal dilatation of $f$ has a lower bound
in terms of the distance of $x$ and $f(x)$ in the distance ratio metric. For
instance, convex domains, bounded domains and domains with uniformly perfect
boundaries are studied.
