On extremal elements and the cardinality of the set of continuously differentiable convex extensions of a Boolean function

In this paper we study the existence of the maximal and minimal elements of the set of continuously differentiable convex extensions to $[0,1]^n$ of an arbitrary Boolean function $f_{B}(x_1,x_2,\ldots,x_n)$ and the cardinality of the set of continuously differentiable convex extensions to $[0,1]^n$ of the Boolean function $f_{B}(x_1,x_2,\ldots,x_n)$. As a result of the study, it was established that the cardinality of the set of continuously differentiable convex extensions to $[0,1]^n$ of an arbitrary Boolean function $f_{B}(x_1,x_2,\ldots,x_n)$ is equal to the continuum. It is argued that for any Boolean function $f_{B}(x_1,x_2,\ldots,x_n)$, there is no minimal element among its continuously differentiable convex extensions to $[0,1]^n$. It is proved that for any Boolean function $f_{B}(x_1,x_2,\ldots,x_n)$, the set of its continuously differentiable convex extensions to $[0,1]^n$ has a maximal element only if the number of essential variables of the given Boolean function $f_{B}(x_1,x_2,\ldots,x_n)$ is less than $2$.