### An optimization problem for continuous submodular functions

#### Abstract

Motivated by problems concerning the Shannon complexity of multipartite secret sharing, a special case of the following general optimization problem is considered: find the minimal cost of those EL functions which satisfy certain constraints.

In our special case the cost of an EL function is the maximal value of the $n$ partial derivatives at zero. Another possibility could be the supremum of the function range. The constraints are specified by a smooth bounded surface $S$ cutting off a downward closed subset. An EL function is feasible if at the internal points of $S$ the left and right partial derivatives of the function differ by at least one.

A general lower bound for the minimal cost is given in terms of the normals of the surface $S$. The bound is tight when $S$ is linear. In the two-dimensional case the same bound is tight for convex or concave $S$. It is shown that the optimal EL function is not necessarily unique. The paper concludes with several open problems.

#### Keywords

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DOI: http://dx.doi.org/10.24193/subbmath.2021.1.17

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