Abstract

We study the class of compact convex subsets of a topological vector space which admit a strictly convex and lower semicontinuous function. We prove that such a compact set is embeddable in a strictly convex dual Banach space endowed with its weak∗^* topology. In addition, we find many exposed points where a strictly convex lower semicontinuous function is continuous. As a consequence, every set in the above class is the closed convex hull of its exposed points.

Author information

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

Luis Carlos García-Lirola

Dep. de Matemáticas, Facultad de Matemáticas, Universidad de Murcia, 30100 Espinardo-Murcia, Spain

luiscarlos.garcia@um.es

José Orihuela

Dep. de Matemáticas, Facultad de Matemáticas, Universidad de Murcia, 30100 Espinardo-Murcia, Spain

joseori@um.es

Matías Raja

Dep. de Matemáticas, Facultad de Matemáticas, Universidad de Murcia, 30100 Espinardo-Murcia, Spain

matias@um.es

Keywords

  • Convex compact set
  • convex lower semicontinuous function
  • exposed point
  • continuity point

Mathematics Subject Classification

46A55; 46B03,54E35

References

  1. R. D. Bourgin: Geometric Aspects of Convex Sets with Radon–Nikodým Property, Lect. Notes in Math. 993, Springer, Berlin (1980).Google Scholar
  2. R. Deville, G. Godefroy, V. Zizler: Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64, Longman Scientific & Technical, Harlow (1993).Google Scholar
  3. R. Deville, V.E. Zizler: Farthest points in w∗ -compact sets, Bull. Aust. Math. Soc. 38(3) (1988) 433–439.https://doi.org/10.1017/S0004972700027817Google Scholar
  4. M. Fabian, P. Habala, P. Hájek, V. Montesinos, J. Pelant, V. Zizler: Functional Analysis and Infinite-Dimensional Geometry, CMS Books in Mathematics 8, Springer, New York (2001).https://doi.org/10.1007/978-1-4757-3480-5Google Scholar
  5. G. Godefroy, D. Li: Strictly convex functions on compact convex sets and applications, in: Functional Analysis, P. K. Jain (ed.), Narosa, New Delhi (1998) 182–192.Google Scholar
  6. M. Hervé: Sur les représentations intégrales à l’aide des points extrémaux dans un ensemble compact convexe métrisable, C. R. Acad. Sci., Paris 253 (1961) 366–368.Google Scholar
  7. S. Ferrari, J. Orihuela, M. Raja: Weakly metrizabilty of spheres and renorming of normed spaces, Q. J. Math. 67(1) (2016) 15–27.https://doi.org/10.1093/qmath/hav038Google Scholar
  8. G. Lancien: On the Szlenk index and the weak*-dentability index, Q. J. Math., Oxf. II. Ser. 47(185) (1996) 59–71.https://doi.org/10.1093/qjmath/47.185.59Google Scholar
  9. A. Moltó, J. Orihuela, S. Troyanski, M. Valdivia: On weakly locally uniformly rotund Banach spaces, J. Funct. Anal. 163(2) (1999) 252–271.https://doi.org/10.1006/jfan.1998.3376Google Scholar
  10. K. F. Ng: On a theorem of Dixmier, Math. Scand. 29 (1971) 279–280.https://doi.org/10.7146/math.scand.a-11054Google Scholar
  11. J. Orihuela, R. Smith, S. Troyanski: Strictly convex norms and topology, Proc. London Math. Soc. (3) 104 (2012) 197–222.https://doi.org/10.1112/plms/pdr034Google Scholar
  12. L. Oncina, M. Raja: Descriptive compact spaces and renorming, Stud. Math. 165(1) (2004) 39–52.https://doi.org/10.4064/sm165-1-3Google Scholar
  13. R. R. Phelps: Convex Functions, Monotone Operators and Differentiability, Lect. Notes in Math. 1364, Springer, Berlin (1993).Google Scholar
  14. M. Raja: Continuity at the extreme points, J. Math. Anal. Appl. 350(2) (2009) 436–438.https://doi.org/10.1016/j.jmaa.2007.11.043Google Scholar
  15. M. Raja: Compact spaces of Szlenk index ω, J. Math. Anal. Appl. 391(2) (2012) 496–509.https://doi.org/10.1016/j.jmaa.2012.02.011Google Scholar
  16. N. K. Ribarska: Internal characterization of fragmentable spaces, Mathematika 34(2) (1987) 243–257.https://doi.org/10.1112/S0025579300013498Google Scholar
  17. N. K. Ribarska: A note on fragmentability of some topological spaces, C. R. Acad. Bulg. Sci. 43(7) (1990) 13–15.Google Scholar
  18. R. Smith: Strictly convex norms, Gδ -diagonals and non-Gruenhage spaces, Proc. Amer. Math. Soc. 140(9) (2012) 3117–3125.https://doi.org/10.1090/S0002-9939-2012-11142-6Google Scholar
  19. C. Stegall: The topology of certain spaces of measures, Topology Appl. 41(1–2) (1991) 73–112.https://doi.org/10.1016/0166-8641(91)90102-RGoogle Scholar

Suggested citation

L. C. García-Lirola, J. Orihuela, M. Raja. “Convex Compact Sets that Admit a Lower Semicontinuous Strictly Convex Function.” Journal of Convex Analysis 24 (2017), No. 3, 987–998. https://doi.org/10.68381/jca24059

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2017. Rights now held by Banach Press LLC.