On the Existence and Properties of Convex Extensions of Boolean Functions

Abstract

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 \([0,1]^n\).