On the order of smoothness of the smallest concave extension of a Boolean function

In this paper, we study the order of smoothness of $f_{NR}(x_1,x_2,\ldots ,x_n)$ — the least concave extension on $[0,1]^n$ of an arbitrary Boolean function $f_B(x_1,x_2,\ldots ,x_n)$. We prove that if the Boolean function $f_B(x_1,x_2,\ldots ,x_n)$ essentially depends on at most one variable, then on $[0,1]^n$ its least concave extension $f_{NR}(x_1,x_2,\ldots ,x_n)$ is infinitely differentiable, otherwise the extension $f_{NR}(x_1,x_2,\ldots ,x_n)$ on $[0,1]^n$ is only continuous. We demonstrate how the least concave extension can be used to solve systems of Boolean equations.