On the construction of bodies of constant width containing a given set

Abstract

A survey of the author’s results is presented, and the research is continued of the known differential geometry problem on the construction of bodies of constant width containing an arbitrary given bounded set of the same diameter as the width of the bodies. The problem is considered for sets from reflexive Banach spaces in which the unit ball is a generating set. Using earlier results in strongly convex analysis and an explicit formula for constructing one of such bodies of constant width, we establish a criterion for the uniqueness of the complement of an arbitrary set to a body of constant width and propose some algorithms for constructing all the bodies of constant width that contain a given set.