Extension of convex functions from a hyperplane to a half-space

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 ${\mathbb{R}}^n\times [0,\mathrm{\infty })$ which is finite and smooth on the open half-space ${\mathbb{R}}^n\times (0,\mathrm{\infty })$. 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 $\psi (A)\to \mathrm{\infty }$ as $\det A\to 0+$.