  It is shown that a possibly infinite-valued proper lower semicontinuous
convex function on ${\mathbb R}^n$ has an extension to a convex function on the
half-space ${\mathbb R}^n\times[0,\infty)$ which is finite and smooth on the
open half-space ${\mathbb R}^n\times(0,\infty)$. The result is applied to
nonlinear elasticity, where it clarifies how the condition of polyconvexity of
the free-energy density $\psi(Dy)$ is best expressed when $\psi(A)\to\infty$ as
$\det A\to 0+$.
