Abstract

We consider the problem of H-separability for two H-convex subsets A and B of Rn\mathbb{R}^n. There are two types of H-separability. The first one, called "strict H-separability", is the separation (in the usual sense) of the sets A and B by an H-convex hyperplane. The second one ("weak H-separability") means to look for an H-convex half-space P such that A is situated in P, whereas B has no point in common with the interior of P. We give necessary and sufficient conditions for both these types of H-separability; the results are connected to the H-convexity of the Minkowski sum of H-convex sets investigated in a previous paper of the authors [J. Combin. Theory, Ser. A 103 (2003) 323–336]. Some examples illustrate the obtained results.

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Keywords

  • Convex sets
  • H-convexity
  • Minkowski addition
  • separation theory

Mathematics Subject Classification

52A01; 52A20

References

  1. V. Boltyanski: On certain classes of convex sets, Soviet Math. Doklady 17(1) (1976) 10–13.Google Scholar
  2. V. Boltyanski: Helly’s theorem for H-convex sets, Soviet Math. Doklady 17(1) (1976) 78–81.Google Scholar
  3. V. Boltyanski: Some theorems of combinatorial geometry (in Russian), Mat. Zametki 21 (1977) 117–124.Google Scholar
  4. V. Boltyanski: The tent method in optimal control theory, in: Optimal Control, International Series of Numerical Mathematics 111, R. Bulirsch et al. (eds.), Birkhäuser, Basel (1993) 3–20.https://doi.org/10.1007/978-3-0348-7539-4_1Google Scholar
  5. V. Boltyanski, H. Martini: On maximal primitive fixing systems, Beitr. Algebra Geom. 37 (1996) 199–207.Google Scholar
  6. V. Boltyanski, H. Martini: Carathéodory’s theorem and H-convexity, J. Combin. Theory, Ser. A 93 (2001) 292–309.https://doi.org/10.1006/jcta.2000.3080Google Scholar
  7. V. Boltyanski, H. Martini: Minkowski addition of H-convex sets and related Helly-type theorem, J. Combin. Theory, Ser. A 103 (2003) 323–336.https://doi.org/10.1016/S0097-3165(03)00101-8Google Scholar
  8. V. Boltyanski, H. Martini, P. Soltan: Excursions into Combinatorial Geometry, Springer, Berlin (1997).https://doi.org/10.1007/978-3-642-59237-9Google Scholar
  9. V. Boltyanski, E. Morales Amaya: Minimal fixing systems for convex bodies, J. Appl. Anal. 1(1) (1995) 1–13.https://doi.org/10.1515/JAA.1995.1Google Scholar
  10. H. G. Eggleston: Convexity, Cambridge University Press, Cambridge (1958).https://doi.org/10.1017/CBO9780511566172Google Scholar
  11. S. R. Lay: Convex Sets and their Applications, J. Wiley & Sons, New York (1982).Google Scholar
  12. K. Leichtweiss: Konvexe Mengen, Deutscher Verlag der Wissenschaften, Berlin (1980).https://doi.org/10.1007/978-3-642-95335-4Google Scholar
  13. F. Nožička, L. Grygarov, K. Lommatzsch: Geometrie konvexer Mengen und konvexe Analysis, Akademie-Verlag, Berlin (1988).https://doi.org/10.1515/9783112727553Google Scholar
  14. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  15. R. Schneider: Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge (1993).https://doi.org/10.1017/CBO9780511526282Google Scholar
  16. J. Stoer, C. Witzgall: Convexity and Optimization in Finite Dimensions I, Springer, Berlin (1970).https://doi.org/10.1007/978-3-642-46216-0Google Scholar
  17. F. A. Valentine: Convex Sets, McGraw-Hill, New York (1964).Google Scholar
  18. R. Webster: Convexity, Oxford University Press, Oxford (1994).https://doi.org/10.1093/oso/9780198531470.001.0001Google Scholar

Suggested citation

V. Boltyanski, H. Martini. “Separability of H-convex Sets.” Journal of Convex Analysis 12 (2005), No. 1, 131–137. https://doi.org/10.68381/jca12008

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Published by Heldermann Verlag, 2005. Rights now held by Banach Press.