Compensated Convexity in a Theorem of H. M. Reimann

It is proved that if 0 < λ ≤ f(x) ≤ Λ for x ∈ Ω, where Ω ⊂ R is a bounded convex domain, and f is L-Dini continuous on Ω, then there exist infinitely many biLipschitz maps F : Ω → R such that detDF (x) = f(x) for a.e. x ∈ Ω. Moreover, these mappings can be chosen to have convex potentials. We relate our result to a classical theorem by H. M. Reimann; however, the emphasis is on the novel use of connections between quasiconformal mappings and the Monge-Ampère equation.