×

Convex compact sets that admit a lower semicontinuous strictly convex function. (English) Zbl 1392.46006

The main result of the article is as follows: Let \(X\) be a locally convex topological vector space and let \(K\subset X\) be a compact convex subset. Then there exists a lower semicontinuous strictly convex real functionon \(K\) if and only if \(K\) embeds linearly into a strictly convex dual Banach space endowed with the weak\(^*\) topology. An application to exposed points is given.

MSC:

46A55 Convex sets in topological linear spaces; Choquet theory
52A07 Convex sets in topological vector spaces (aspects of convex geometry)
PDFBibTeX XMLCite
Full Text: arXiv Link

References:

[1] R. D. Bourgin: Geometric Aspects of Convex Sets with Radon-Nikod´ym Prop-erty, Lect. Notes in Math. 993, Springer, Berlin (1980).
[2] R. Deville, G. Godefroy, V. Zizler: Smoothness and Renormings in BanachSpaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64,Longman Scientific & Technical, Harlow (1993). · Zbl 0782.46019
[3] R. Deville, V.E. Zizler: Farthest points in w∗-compact sets, Bull. Aust. Math.Soc. 38(3) (1988) 433-439. · Zbl 0656.46012
[4] M. Fabian, P. Habala, P. H´ajek, V. Montesinos, J. Pelant, V. Zizler: FunctionalAnalysis and Infinite-Dimensional Geometry, CMS Books in Mathematics 8,Springer, New York (2001). · Zbl 0981.46001
[5] G. Godefroy, D. Li: Strictly convex functions on compact convex sets and ap-plications, in: Functional Analysis, P. K. Jain (ed.), Narosa, New Delhi (1998)182-192. · Zbl 0943.43001
[6] M. Herv´e: Sur les repr´esentations int´egrales ‘a l’aide des points extr´emaux dansun ensemble compact convexe m´etrisable, C. R. Acad. Sci., Paris 253 (1961)366-368.
[7] S. Ferrari, J. Orihuela, M. Raja: Weakly metrizabilty of spheres and renormingof normed spaces, Q. J. Math. 67(1) (2016) 15-27. · Zbl 1345.46005
[8] G. Lancien: On the Szlenk index and the weak*-dentability index, Q. J. Math.,Oxf. II. Ser. 47(185) (1996) 59-71. · Zbl 0973.46014
[9] A. Molt´o, J. Orihuela, S. Troyanski, M. Valdivia: On weakly locally uniformlyrotund Banach spaces, J. Funct. Anal. 163(2) (1999) 252-271. · Zbl 0927.46010
[10] K. F. Ng: On a theorem of Dixmier, Math. Scand. 29 (1971) 279-280. · Zbl 0243.46023
[11] J. Orihuela, R. Smith, S. Troyanski: Strictly convex norms and topology, Proc.London Math. Soc. (3) 104 (2012) 197-222. · Zbl 1241.46005
[12] L. Oncina, M. Raja: Descriptive compact spaces and renorming, Stud. Math.165(1) (2004) 39-52. · Zbl 1101.46014
[13] R. R. Phelps: Convex Functions, Monotone Operators and Differentiability,Lect. Notes in Math. 1364, Springer, Berlin (1993). · Zbl 0921.46039
[14] M. Raja: Continuity at the extreme points, J. Math. Anal. Appl. 350(2) (2009)436-438. · Zbl 1165.46001
[15] M. Raja: Compact spaces of Szlenk index ω, J. Math. Anal. Appl. 391(2) (2012)496-509. · Zbl 1257.46007
[16] N. K. Ribarska: Internal characterization of fragmentable spaces, Mathematika34(2) (1987) 243-257. · Zbl 0645.46017
[17] N. K. Ribarska: A note on fragmentability of some topological spaces, C. R.Acad. Bulg. Sci. 43(7) (1990) 13-15. · Zbl 0763.54021
[18] R. Smith: Strictly convex norms, Gδ-diagonals and non-Gruenhage spaces, Proc.Amer. Math. Soc. 140(9) (2012) 3117-3125. · Zbl 1282.46015
[19] C. Stegall: The topology of certain spaces of measures, Topology Appl. 41(1-2)(1991) 73-112. · Zbl 0773.46012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.