 
Summary: Computational Aspects of Helly's Theorem and its Relatives
David Avis* and Michael E. Houle**
*School of Computer Science, McGill University
Montr'eal, Qu'ebec, Canada H3A 2A7
**Department of Computer Science, University of Newcastle,
Newcastle, New South Wales 2308, Australia
Abstract
This paper investigates computational aspects of the wellknown convexity theorem due to
Helly, which states that the existence of a point in the common intersection of n convex sets is
guaranteed by the existence of points in the common intersection of each combination of d + 1
of these sets. Given an oracle which accepts d + 1 convex sets and either returns a point in
their common intersection, or reports its nonexistence, we give two algorithms which compute a
point in the common intersection of n such sets. The first algorithm runs in O(n d+1 T ) time and
O(n d ) space, where T is the time required for a single call to the oracle. The second algorithm
is a multistage variant of the first by which the space complexity may be reduced to O(n) at
the expense of an increase in the time complexity by a factor independent of n.
We also show how these algorithms may be adapted to construct linear and spherical sepa
rators of a collection of sets, and to construct a translate of a given object which either contains,
is contained by, or intersects a collection of convex sets.
1 Introduction
