  A relatively polynomially convex subset $V$ of a domain $\Omega$ has the
extension property if for every polynomial $p$ there is a bounded holomorphic
function $\phi$ on $\Omega$ that agrees with $p$ on $V$ and whose $H^\infty$
norm on $\Omega$ equals the sup-norm of $p$ on $V$. We show that if $\Omega$ is
either strictly convex or strongly linearly convex in ${\mathbb C}^2$, or the
ball in any dimension, then the only sets that have the extension property are
retracts. If $\Omega$ is strongly linearly convex in any dimension and $V$ has
the extension property, we show that $V$ is a totally geodesic submanifold. We
show how the extension property is related to spectral sets.
