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