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SEPARATION OF CONVEX SETS IN EXTENDED NORMED SPACES

  • G. BEER (a1) and J. VANDERWERFF (a2)

Abstract

We give continuous separation theorems for convex sets in a real linear space equipped with a norm that can assume the value infinity. In such a space, it may be impossible to continuously strongly separate a point $p$ from a closed convex set not containing $p$ , that is, closed convex sets need not be weakly closed. As a special case, separation in finite-dimensional extended normed spaces is considered at the outset.

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[1]Beer, G., ‘The structure of extended real-valued metric spaces’, Set-Valued Var. Anal. 21 (2013), 591602.
[2]Beer, G., Costantini, C. and Levi, S., ‘Bornological convergence and shields’, Mediterr. J. Math. 10 (2013), 529560.
[3]Beer, G., ‘Norms with infinite values’, J. Convex Anal. 22 (2015), 3558.
[4]Beer, G. and Hoffman, M., ‘The Lipschitz metric for real-valued continuous functions’, J. Math. Anal. Appl. 406 (2013), 229236.
[5]Beer, G. and Levi, S., ‘Uniform continuity, uniform convergence, and shields’, Set-Valued Var. Anal. 18 (2010), 251275.
[6]Borwein, J. and Vanderwerff, J., Convex Functions: Constructions, Characterizations and Counterexamples, Encyclopedia of Mathematics and its Applications, 109 (Cambridge University Press, Cambridge, 2010).
[7]Fabian, M., Habala, P., Hájek, P., Montesinos, V., Pelant, J. and Zizler, V., Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Mathematics, 8 (Springer, New York, 2001).
[8]Holmes, R., Geometric Functional Analysis and its Applications (Springer, New York, 1975).
[9]Rockafellar, R. T., Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
[10]Valentine, F. A., Convex Sets (McGraw-Hill, New York, 1964).
[11]Zalinescŭ, C., Convex Analysis in General Vector Spaces (World Scientific, River Edge, NJ, 2002).
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SEPARATION OF CONVEX SETS IN EXTENDED NORMED SPACES

  • G. BEER (a1) and J. VANDERWERFF (a2)

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