Some Computational Aspects of Helly-type Theorems

In this paper, we prove that, for a given positive number d, if every n + 1 of a collection of compact convex sets in IE 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/ √ n if n is odd and d1 = d √ n+ 2/ (n+ 1) if n is even (d1 = 2d− d √ 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 IE 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. c © 2008 European Society of Computational Methods in Sciences and Engineering