Convexity and Teichm\"{u}ller spaces

We provide restricted negative answers to the Royden-Sullivan problem
whether any Teichm\"{u}ller space of dimension greater than $1$
is biholomorphically equivalent to bounded domain in a complex Banach
space. The only known result here is Tukia's theorem of 1977 that there is
a real analytic homeomorphism of the universal Teichm\"{u}ller
space onto a convex domain in some Banach space.
   We prove:
(a) Any Teichm\"{u}ller space $\mathbf T(0,n)$ of the punctured spheres
(the surfaces of genus zero) with sufficiently large number of punctures
$(n \ge n_0 > 4)$ cannot be mapped biholomorphically onto a bounded
convex domain in $\mathbf C^{n-3}$.
(b) The universal Teichm\"{u}ller space is not biholomorphically equivalent
to a bounded convex domain in uniformly convex Banach space, in
particular, to convex domain in the Hilbert space.
  The proofs involve the existence of conformally rigid domains established
by Thurston and some interpolation results for bounded univalent functions.