Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-25T05:35:03.237Z Has data issue: false hasContentIssue false
  • English
  • Français

Spira's theorems on complete linear proofs of systems of linear inequalities

Published online by Cambridge University Press:  26 February 2010

Victor Klee
Victor Klee
Affiliation:
Department of Mathematics, The University of Washington, Seattle, WA 98195, U.S.A.
Get access

Extract

Motivated by questions of computational complexity, Rabin [7] introduced the notion of a complete proof of a system of inequalities. His work and the related paper of Spira [8] should interest geometers as well as computer scientists, for both papers involve convexity in an essential way. Spira's results concern the possibility of covering the intersection of a convex set C and a convex polyhedron Q with a finite collection P of polyhedra subject to certain conditions, while in Rabin's work the members of P may be more general than polyhedra. Both papers are interesting and treat important questions, but only Rabin's paper is correct in all respects. The present note contains counterexamples to some of Spira's results and establishes a correct version of one of them.

Type
Research Article
Information
Mathematika , Volume 22 , Issue 2 , December 1975 , pp. 112 - 121
Copyright
Copyright © University College London 1975

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Goldman, A. J.. “Resolution and separation theorems for polyhedral convex sets, linear inequalities and related systems”, Annals of Mathematics Studies, no. 38, 41–51 (Princeton Univ. Press, Princeton, New Jersey, 1956).Google Scholar
2.Goldman, A. J. and Tucker, A. W.. “Polyhedral convex cones, linear inequalities and related systems”, Annals of Mathematics Studies, no. 38, 19–40 (Princeton Univ. Press, Princeton, New Jersey, 1956).Google Scholar
3.Griinbaum, B.. Convex poly topes (Wiley, New York, 1967).Google Scholar
4.Klee, V.. “Extremal structure of convex sets”, Archiv der Math., 8 (1957), 234–240.CrossRefGoogle Scholar
5.Klee, V.. “Some characterizations of convex polyhedra”, Acta Math., 102 (1959), 79–107.CrossRefGoogle Scholar
6.Motzkin, Th.. Beitrdge zur Theorie der linearen Ungleichungen, Dissertation (Basel, 1936).Google Scholar
7.Rabin, M. O.. “Proving simultaneous positivity of linear forms”, J. Computer and System Sci., 6 (1972), 639–650.CrossRefGoogle Scholar
8.Spira, P. M.. “Complete linear proofs of systems of linear inequalities”, J. Computer and System Sci., 6 (1972), 205–216.CrossRefGoogle Scholar