Keywords: Helly’s theorem; set of width; set of constant width
Mathematics Subject Classifcation: 52A01
Abstract: In this paper, we prove that, for a given positive number d, if every n + 1 of
a collection of compact convex sets in IEn contain a set of width d (a set of constant
width d, respectively) simultaneously, then all members of this collection contain a set
of constant width d1 simultaneously, where d1 = d=pn if n is odd and d1 = dpn + 2=
(n + 1) if n is even (d1 = 2d ? d
p
2n=(n + 1), respectively). This set is called common set
(of constant width d1) of the collection. These results deal with an open question raised
by Buchman and Valentine in [Croft, Falconer and Guy, Unsolved Problems in Geometry,
Springer-Verlag New York, Inc. 1991, pp. 131-132]. Moreover, given an oracle which
accepts n + 1 sets of a collection of compact convex sets in IEn and either returns a set of
width d (a set of constant width d) contained in these sets, or reports its non-existence,
we present an algorithm which determines a common set of the collection.