## Extreme points of lattice intervals in the Minkowski-Rådström-Hörmander lattice

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- by Jerzy Grzybowski and Ryszard Urbański
- Proc. Amer. Math. Soc.
**136**(2008), 3957-3962 - DOI: https://doi.org/10.1090/S0002-9939-08-09376-3
- Published electronically: June 3, 2008

## Abstract:

In this paper we characterize extreme points of any symmetric interval in the Minkowski–Rådström–Hörmander lattice $\widetilde {X}$ over any Hausdorff topological vector space $X$ (Theorem 1). Then we prove that the unit ball in the Minkowski–Rådström–Hörmander lattice $\widetilde {X}$ over any normed space $X$, dim$X\geq 2,$ has exactly two extreme points (Theorem 2).## References

- Garrett Birkhoff,
*Lattice theory*, 3rd ed., American Mathematical Society Colloquium Publications, Vol. 25, American Mathematical Society, Providence, R.I., 1979. MR**598630** - Frank H. Clarke,
*Optimization and nonsmooth analysis*, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR**709590** - Romulus Cristescu,
*Topological vector spaces*, Editura Academiei, Bucharest; Noordhoff International Publishing, Leyden, 1977. Translated from the Romanian by Mihaela Suliciu. MR**0454552** - Gerard Debreu,
*Integration of correspondences*, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 351–372. MR**0228252** - V. F. Demyanov and A. M. Rubinov,
*An introduction to quasidifferential calculus*, Quasidifferentiability and related topics, Nonconvex Optim. Appl., vol. 43, Kluwer Acad. Publ., Dordrecht, 2000, pp. 1–31. MR**1766791**, DOI 10.1007/978-1-4757-3137-8_{1} - L. Drewnowski,
*Additive and countably additive correspondences*, Comment. Math. Prace Mat.**19**(1976), no. 1, 25–54. MR**422564** - Pierre Goossens,
*Completeness of spaces of closed bounded convex sets*, J. Math. Anal. Appl.**115**(1986), no. 1, 192–201. MR**835594**, DOI 10.1016/0022-247X(86)90033-8 - J. Grzybowski and R. Urbański,
*On inclusion and summands of bounded closed convex sets*, Acta Math. Hungar.**106**(2005), no. 4, 293–300. MR**2131334**, DOI 10.1007/s10474-005-0020-6 - Lars Hörmander,
*Sur la fonction d’appui des ensembles convexes dans un espace localement convexe*, Ark. Mat.**3**(1955), 181–186 (French). MR**68112**, DOI 10.1007/BF02589354 - Diethard Pallaschke and Ryszard Urbański,
*Pairs of compact convex sets*, Mathematics and its Applications, vol. 548, Kluwer Academic Publishers, Dordrecht, 2002. Fractional arithmetic with convex sets. MR**1961230**, DOI 10.1007/978-94-015-9920-7 - Prakash Prem and Murat R. Sertel,
*Hyperspaces of topological vector spaces: their embedding in topological vector spaces*, Proc. Amer. Math. Soc.**61**(1976), no. 1, 163–168 (1977). MR**425881**, DOI 10.1090/S0002-9939-1976-0425881-3 - Hans Rådström,
*An embedding theorem for spaces of convex sets*, Proc. Amer. Math. Soc.**3**(1952), 165–169. MR**45938**, DOI 10.1090/S0002-9939-1952-0045938-2 - Stefan Scholtes,
*Minimal pairs of convex bodies in two dimensions*, Mathematika**39**(1992), no. 2, 267–273. MR**1203283**, DOI 10.1112/S002557930001500X - R. Urbański,
*A generalization of the Minkowski-Rȧdström-Hörmander theorem*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.**24**(1976), no. 9, 709–715 (English, with Russian summary). MR**442646**

## Bibliographic Information

**Jerzy Grzybowski**- Affiliation: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland
- Email: jgrz@amu.edu.pl
**Ryszard Urbański**- Affiliation: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland
- Email: rich@amu.edu.pl
- Received by editor(s): March 13, 2007
- Received by editor(s) in revised form: October 4, 2007
- Published electronically: June 3, 2008
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2008 Jerzy Grzybowski and Ryszard Urbański
- Journal: Proc. Amer. Math. Soc.
**136**(2008), 3957-3962 - MSC (2000): Primary 46B20, 52A05, 54B20
- DOI: https://doi.org/10.1090/S0002-9939-08-09376-3
- MathSciNet review: 2425736