Subsets in a Boolean cube layer which are minimal by double-sided shadow and distinct from disks

Abstract

We consider the double-shadow minimization problem in a Boolean cube layer. It is proved that a right lexicographic segment of the second layer has the minimal double-sided shadow. The minimal families of size 1 + k(n − k) + (k − 1)(n − k − 1) in the kth layer are described when n = 2k and for small values of k.