  Given a finite sequence $a:={a_1, ..., a_N}$ in a domain $\Omega \subset
C^n$, and complex scalars $v:={v_1, ..., v_N}$, consider the classical extremal
problem of finding the smallest uniform norm of a holomorphic function
verifying $f(a_j)=v_j$ for all $j$. We show that the modulus of the solutions
to this problem must approach its least upper bound along a subset of the
boundary of the domain large enough to contain the support of a measure whose
hull contains a subset of the original $a$ large enough to force the same
minimum norm. Furthermore, all the solutions must agree on a variety which also
contains this hull. An example is given to show that the inclusions can be
strict.
