  Quasiconformal homeomorphisms of the whole space Rn, onto itself normalized
at one or two points are studied. In particular, the stability theory, the case
when the maximal dilatation tends to 1, is in the focus. Our main result
provides a spatial analogue of a classical result due to Teichm\"uller. Unlike
Teichm\"uller's result, our bounds are explicit. Explicit bounds are based on
two sharp well-known distortion results: the quasiconformal Schwarz lemma and
the bound for linear dilatation. Moreover, Bernoulli type inequalities and
asymptotically sharp bounds for special functions involving complete elliptic
integrals are applied to simplify the computations. Finally, we discuss the
behavior of the quasihyperbolic metric under quasiconformal maps and prove a
sharp result for quasiconformal maps of R^n \ {0} onto itself.
