Theory of locally concave functions and its applications to sharp   estimates of integral functionals
  We prove a duality theorem the computation of certain Bellman functions is usually based on. As a byproduct, we obtain sharp results about the norms of monotonic rearrangements. The main novelty of our approach is a special class of martingales and an extremal problem on this class, which is dual to the minimization problem for locally concave functions. 
Dmitriy M. Stolyarov, Pavel B. Zatitskiy
math.OC math.CA math.PR