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