On the Existence and Properties of Convex Extensions of Boolean Functions
Dostonjon Numonjonovich Barotov · Mathematical Notes · 2024
We study the problem of the existence of a convex extension of any Boolean function $$f(x_1,x_2,\dots,x_n)$$ to the set $$[0,1]^n$$ . A convex extension $$f_C(x_1,x_2,\dots,x_n)$$ of an arbitrary Boolean function $$f(x_1,x_2,\dots,x_n)$$ to the set $$[0,1]^n$$ is constructed. On the basis of the constructed convex extension $$f_C(x_1,x_2,\dots,x_n)$$ , it is proved that any Boolean function $$f(x_1,x_2,\dots,x_n)$$ has infinitely many convex extensions to $$[0,1]^n$$ . Moreover, it is proved constructively that, for any Boolean function $$f(x_1,x_2,\dots,x_n)$$ , there exists a unique function $$f_{DM}(x_1,x_2,\dots,x_n)$$ being its maximal convex extensions to $$[0,1]^n$$ .