R U {+oo}, we have (i) rcf
Envelop. Since the pointwise supremum of convex functions is clearly convex, Cf is convex. Moreover Theorem 2 reads as Theorem 4. Let f : V -+ IR U {+oo} and n = dim V. Then we have n+1 n+1 Cf (x) = inf { \if (xi) i=1 I l Ai > 0, E Ai = 1, xi E V } i=1 . c.. In this case the epigraph of f epif :_ {(x,y)EVxRjy? f(x)} is convex and closed. Therefore Hahn-Banach's theorem yields Proposition 2. c.. Then f is never -oo if and only if there exists an affine map P : V -+ 1R such that $(x) < f (x) V x E V.
Regular Variational Integrals 8 Proof. Denote by LT (x) Lebesgue's extension Lr(x) := inf{l f f(xk) I {xk} C Y, Xk - x} . , that the set Ct :_ {xEXILT(x)