  Polynomial spaces associated to a convex body $C$ in $({\bf R}^+)^d$ have
been the object of recent studies. In this work, we consider polynomial spaces
associated to non-convex $C$. We develop some basic pluripotential theory
including notions of $C-$extremal plurisubharmonic functions $V_{C,K}$ for
$K\subset {\bf C}^d$ compact. Using this, we discuss Bernstein-Walsh type
polynomial approximation results and asymptotics of random polynomials in this
non-convex setting.
