## Illinois Journal of Mathematics

### On the Krein–Milman–Ky Fan theorem for convex compact metrizable sets

Mohammed Bachir

#### Abstract

We extend the extension by Ky Fan of the Krein–Milman theorem. The $\Phi$-extreme points of a $\Phi$-convex compact metrizable space are replaced by the $\Phi$-exposed points in the statement of Ky Fan theorem. Our main results are based on new variational principles which are of independent interest. Several applications will be given.

#### Note

Due to computer-generated errors that were introduced in the typesetting stage, this article, which originally appeared in the Illinois Journal of Mathematics (Volume 61:1–2, Spring and Summer 2017), is being reprinted in its entirety. The publisher apologizes for any inconvenience to readers.

#### Article information

Source
Illinois J. Math., Volume 62, Number 1-4 (2018), 1-24.

Dates
Revised: 2 September 2017
First available in Project Euclid: 13 March 2019

https://projecteuclid.org/euclid.ijm/1552442654

Digital Object Identifier
doi:10.1215/ijm/1552442654

Mathematical Reviews number (MathSciNet)
MR3922408

Zentralblatt MATH identifier
07036778

#### Citation

Bachir, Mohammed. On the Krein–Milman–Ky Fan theorem for convex compact metrizable sets. Illinois J. Math. 62 (2018), no. 1-4, 1--24. doi:10.1215/ijm/1552442654. https://projecteuclid.org/euclid.ijm/1552442654

#### References

• J. Araujo and J. J. Font, On Shilov boundaries for subspaces of continuous functions, Topology Appl. 77 (1997), 79–85.
• J. Araujo and J. J. Font, Linear isometries between subspaces of continuous functions, Trans. Amer. Math. Soc. 349 (1997), 413–428.
• E. Asplund and R. T. Rockafellar, Gradients of convex functions, Trans. Amer. Math. Soc. 139 (1969), 443–467.
• M. Bachir, A non convexe analogue to Fenchel duality, J. Funct. Anal. 181 (2001), 300–312.
• M. Bachir, Limited operators and differentiability, North-West. Eur. J. Math. 3 (2017), 63–73.
• M. Bachir, An extension of the Banach–Stone theorem, J. Aust. Math. Soc. 105 (2018), 1–23.
• H. S. Bear, The Shilov boundary for a linear space of continuous functions, Amer. Math. Monthly 68 (1961), 483–485.
• J. Bourgain, Strongly exposed points in weakly compact convex sets in Banach spaces, Proc. Amer. Math. Soc. 58 (1976), 197–200.
• B. Cascales and J. Orihuela, On compactness in locally convex spaces, Math. Z. 195 (1987), 365–381.
• R. Deville, G. Godefroy and V. Zizler, A smooth variational principle with applications to Hamilton–Jacobi equations in infinite dimensions, J. Funct. Anal. 111 (1993), 197–212.
• R. Deville and J. P. Revalski, Porosity of ill-posed problems, Proc. Amer. Math. Soc. 128 (2000), 1117–1124.
• K. Fan, On the Krein–Milman theorem, Proceedings of symposia in pure mathematics, vol. 7, Amer. Math. Soc., Providence, RI, 1963, pp. 211–219.
• P. Habala, P. Hàjek and V. Zizler, Introduction to Banach spaces, Lecture Notes in Mathematics, Matfyzpress, Charles University, Prague, 1996.
• B. D. Khanh, Sur la $\Phi$-Convexité de Ky Fan, J. Math. Anal. Appl. 20 (1967), 188–193.
• V. Klee, Extremal structure of convex sets II, Math. Z. 69 (1958), 90–104.
• M. Krein and D. Milman, On extreme points of regular convex sets, Studia Math. 9 (1940), 133–138.
• D. P. Milman, Characteristic of extremal functions, Dokl. Akad. Nauk SSSR 57 (1947), 119–122. (In Russian.)
• D. P. Milman, On integral representations of functions of several variables, Dokl. Akad. Nauk SSSR 87 (1952), 9–10. (In Russian.)
• R. R. Phelps, Convex functions, monotone operators and differentiability, Lecture Notes in Mathematics, vol. 1364, Springer-Verlag, Berlin, 1993.
• R. R. Phelps, Lectures on Choquet's theorem, 2nd ed., Lecture Notes in Mathematics, vol. 1757, Springer, Berlin, 1997.
• J. W. Roberts, A compact convex set with no extreme points, Studia Math. 60 (1977), no. 3, 255–266.
• A. E. Taylor and D. C. Lay, Introduction to functional analysis, 2nd ed., Wiley, New York, 1980.
• L. Zajicek, On the differentiation of convex functions in finite and infinite dimensional spaces, Czechoslovak Math. J. 29 (1979), no. 3, 340–348.
• L. Zajicek, On sigma-porous sets in abstract spaces, Abstr. Appl. Anal. 2005 (2005), no. 5, 509–534.