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We investigate the minima of functionals of the form $$\int_{[a,b]}g(\dot u(s)){\rm d}s$$ 
where g is strictly convex. The admissible functions $u:[a,b]\longrightarrow\mathbb{R}$ 
are not necessarily convex and satisfy $u\leq f$ 
on [a,b], u(a)=f(a), u(b)=f(b), f is a fixed function on [a,b].We show that the minimum is attained by $\bar f$ 
, the convex envelope of f.