It is shown that a possibly infinite-valued proper lower semicontinuous convex function on R n has an extension to a convex function on the half-space R n × [ 0 , ∞) which is finite and smooth on the open half-space R n × (0 , ∞) . The result is applied to nonlinear elasticity, where it clarifies how the condition of polyconvexity of the free-energy density ψ (D y) is best expressed when ψ (A) → ∞ as det A → 0 + .