  Let $\Omega$ be a domain in $R^n$, and let $N=3\cdot 2^{n-1}$. We prove that
the trace of the space $C^2(\Omega)$ to the boundary of $\Omega$ has the
following finiteness property: A function $f:\partial\Omega\to R$ is the trace
to the boundary of a function $F\in C^2(\Omega)$ provided there exists a
constant $\lambda>0$ such that for every set $E\subset\partial\Omega$
consisting of at most $N$ points there exists a function $F_E\in C^2(\Omega)$
with $\|F_E\|_{C^2(\Omega)}\le\lambda$ whose trace to $\partial\Omega$
coincides with $f$ on $E$. We also prove a refinement of this finiteness
principle, which shows that in this criterion we can use only $N$-point subsets
$E\subset\partial\Omega$ which have some additional geometric ``visibility''
properties with respect to the domain $\Omega$.
