On the set of continuously differentiable concave extensions of a Boolean function

This paper is devoted to the study of the existence of extremal elements of the set of continuously differentiable concave extensions to the set $[0,1]^n$ of an arbitrary Boolean function $f_{B}(x_1,\ldots,x_n)$, as well as finding the cardinality of the set of continuously differentiable concave extensions to $[0,1]^n$ of the Boolean function $f_{B}(x_1,\ldots,x_n).$ As a result of the study, it is proved that, firstly, for any Boolean function $f_{B}(x_1,\ldots,x_n)$ among its continuously differentiable concave extensions to $[0,1]^n$ there is no maximal element, secondly, if the Boolean function $f_{B}(x_1,\ldots,x_n)$ has more than one essential variable, then among its continuously differentiable concave extensions to $[0,1]^n$ there is no minimal element, and if the Boolean function is constant or has only one essential variable, then among its continuously differentiable concave extensions to $[0,1]^n$ there is a unique minimal element, the explicit form of which is given in the paper. It was also established that the cardinality of the set of continuously differentiable concave extensions to $[0,1]^n$ of an arbitrary Boolean function $f_{B}(x_1,\ldots,x_n)$ is equal to the continuum.