Abstract

Combining existing approaches, we provide a uniform way to describe all hyperplanes which separate (properly, or strong\-ly) a given pair of non\-emp\-ty convex sets K1K_1 and K2K_2 in the nn-dimensional Euclidean space. The method is based on considering (n ⁣− ⁣1)(n\! -\! 1)-dimensional subspaces which bound the set K1 ⁣− ⁣K2K_1 \!-\! K_2 and then using properties of the polar cone (K1 ⁣− ⁣K2)∘(K_1 \!-\! K_2)^\circ. First, we characterize all separating hyperplanes with given normal vectors, and then those containing a given point. We also describe the union of all hyperplanes separating (properly separating) a given pair of convex sets.

Author information

Contact details are reproduced from the original publication and may be historical.

Valeriu Soltan

Dept. of Mathematical Sciences, George Mason University, Fairfax, VA 22030, U.S.A.

vsoltan@gmu.edu

Keywords

  • Separation
  • hyperplane
  • convex
  • cone

Mathematics Subject Classification

52A20, 90C25

References

  1. H. Brunn: Sätze über zwei getrennte Eikörper, Math. Ann. 104 (1931) 300–324.https://doi.org/10.1007/BF01457938Google Scholar
  2. C. Carathéodory: Über den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen, Rend. Circ. Mat. Palermo 32 (1911) 193–217.https://doi.org/10.1007/BF03014795Google Scholar
  3. Z. R. Gabidullina: A theorem on strict separability of convex polyhedra and its applications in optimization, J. Optim. Theory Appl. 148 (2011) 550–570.https://doi.org/10.1007/s10957-010-9767-1Google Scholar
  4. A. I. Golikov, Yu. G. Evtushenko, S. Ketabchi: On families of hyperplanes that separate polyhedra, Comput. Math. Math. Phys. 45 (2005) 227–242.Google Scholar
  5. L. Grygarová: A calculation of all separating hyperplanes of two convex polytopes, Optimization 41 (1997) 57–69.https://doi.org/10.1080/02331939708844325Google Scholar
  6. L. Grygarová: On a calculation of an arbitrary separating hyperplane of convex polyhedral sets, Optimization 43 (1998) 93–112.https://doi.org/10.1080/02331939808844377Google Scholar
  7. V. L. Klee: Separation and support properties of convex sets – a survey, in: Control Theory and the Calculus of Variations, A. V. Balakrishnan (ed.), Academic Press, New York (1969) 235–303.Google Scholar
  8. J. Lawrence, V. Soltan: On unions and intersections of nested families of cones, Beitr. Algebra Geom. 57 (2016) 655–665.https://doi.org/10.1007/s13366-016-0285-7Google Scholar
  9. H. Minkowski: Geometrie der Zahlen I, Teubner, Leipzig (1896).Google Scholar
  10. H. Minkowski: Gesammelte Abhandlungen, Band 2, Teubner, Leipzig (1911).Google Scholar
  11. B. S. Mordukhovich, N. M. Nam: An Easy Path to Convex Analysis and Applications, Morgan & Claypool, Williston (2014).https://doi.org/10.1007/978-3-031-02406-1Google Scholar
  12. Y. Nievergelt: Parameter spaces of separating hyperplanes, Optimization 64 (2015) 1409–1438.https://doi.org/10.1080/02331934.2013.869812Google Scholar
  13. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  14. V. Soltan: Polarity and separation of cones, Linear Algebra Appl. 538 (2018) 212–224.https://doi.org/10.1016/j.laa.2017.10.019Google Scholar
  15. V. Soltan: Moreau-type characterizations of polar cones, Linear Algebra Appl. 567 (2019) 45–62.https://doi.org/10.1016/j.laa.2019.01.006Google Scholar
  16. V. Soltan: Lectures on Convex Sets, 2nd edition, World Scientific, Hackensack (2020).https://doi.org/10.1142/11328Google Scholar

Suggested citation

V. Soltan. “Separating Hyperplanes of Convex Sets.” Journal of Convex Analysis 28 (2021), No. 4, 1015–1032. https://doi.org/10.68381/jca28059

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Published by Heldermann Verlag, 2021. Rights now held by Banach Press LLC.