Some minimization problems in the class of convex functions with   prescribed determinant
  We consider minimizers of linear functionals of the type $$L(u)=\int_{\p \Omega} u \, d \sigma - \int_{\Omega} u \, dx$$ in the class of convex functions $u$ with prescribed determinant $\det D^2 u =f$.   We obtain compactness properties for such minimizers and discuss their regularity in two dimensions. 
Nam Le, Ovidiu Savin
math.AP