An optimization problem for continuous submodular functions

"Real continuous submodular functions, as a generalization of the corresponding discrete notion to domain, gained considerable attention recently. The analog for entropy functions requires additional properties: real function defined on non-negative orthant $\R^n$ is entropy-like (EL) if it submodular, takes zero at zero, non-decreasing, and has Diminishing Returns property. Motivated by problems concerning Shannon complexity multipartite secret sharing, special case following general optimization problem considered: find minimal cost those EL which satisfy certain constraints. In our an maximal value $n$ partial derivatives zero. Another possibility could be supremum range. constraints are specified smooth bounded surface $S$ cutting off downward closed subset. An feasible internal points left right differ least one. A lower bound given in terms normals $S$. tight when linear. two-dimensional same convex or concave It shown that optimal not necessarily unique. paper concludes with several open problems."