Minimal in terms of double-sided shadow subsets of Boolean cube layer distinct from circles

The double-shadow minimization problem in the Boolean cube layer is considered. The final lexicographical segment of the second layer is shown to have the minimal double-sided shadow. The minimal families of size $1+k(n-k)+(k-1)(n-k-1)$ in the $k$th layer are described when $n=2k$ for small values of $k$. Bibliogr. 5.