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Asymmetric separation of convex sets. (English) Zbl 1454.52005

The paper under review concerns the theory of convex bodies and deals with the separation of convex sets in \(\mathbb R^n\). There exist different ways to define the notion of separation, which are symmetric in the sense that separated sets play equal roles in definitions, see [V. L. Klee, in: Control theory and the calculus of variations. Based on the lectures presented at the workshop on calculus of variations and control theory, University of California, Los Angeles, July 1968. New York-London: Academic Press (1969; Zbl 0203.46901), p. 235–303]. The author introduces various asymmetric notions of separation and then provides a uniform description of existing types of separation.

MSC:

52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)

Citations:

Zbl 0203.46901
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References:

[1] J. Bair, F. Jongmans,La s´eparation vraie dans un espace vectoriel. Bull. Soc. Roy. Sci. Li‘ege 41 (1972), 163-170. · Zbl 0239.46010
[2] A. Brøndsted,The inner aperture of a convex set. Pacific J. Math. 72 (1977), 335-340. · Zbl 0357.52005
[3] V. L. Klee,Strict separation of convex sets. Proc. Amer. Math. Soc. 7 (1956), 735-737. · Zbl 0071.10601
[4] V. L. Klee,Maximal separation theorems for convex sets, Trans. Amer. Math. Soc. 134 (1968), 133-147. · Zbl 0164.52702
[5] V. L. Klee,Separation and support properties of convex sets – a survey, Control Theory and the Calculus of Variations (University of California, LA, 1968), pp. 235-303, Academic Press, New York, 1969.
[6] J. Lawrence,V. Soltan,On unions and intersections of nested families of cones, Beitr. Algebra Geom. 57 (2016), 655-665. · Zbl 1358.52008
[7] H. Minkowski,Geometrie der Zahlen. I, Teubner, Leipzig, 1896; II. Teubner, Leipzig, 1910.
[8] H. Minkowski,Gesammelte Abhandlungen. Bd 2, Teubner, Leipzig, 1911.
[9] R. T. Rockafellar,Convex Analysis. Princeton Universty Press, Princeton, NJ, 1970. · Zbl 0193.18401
[10] V. Soltan,Polarity and separation of cones, Linear Algebra Appl. 538 (2018), 212-224. · Zbl 1374.90308
[11] V. Soltan,Lectures on convex sets. Second edition, World Scientific, Hackensack, NJ, 2020. · Zbl 1431.52001
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