On the Existence and Properties of Convex Extensions of Boolean Functions

We study the problem of the existence of a convex extension of any Boolean function f(x(1), x(2), ..., x(n)) to the set [0, 1](n). A convex extension f(C)(x(1), x(2), ..., x(n)) of an arbitrary Boolean function f(x(1), x(2), ..., x(n)) to the set [0, 1](n) is constructed. On the basis of the constructed convex extension f(C)(x(1), x(2), ..., x(n)), it is proved that any Boolean function f(x(1), x(2), ..., 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), ..., x(n)), there exists a unique function f(DM)(x(1), x(2), ..., x(n)) being its maximal convex extensions to [0, 1](n).