Extension of Simons' inequality

Authors:
Kersti Kivisoo and Eve Oja

Journal:
Proc. Amer. Math. Soc. **133** (2005), 3485-3496

MSC (2000):
Primary 39B62, 46A55, 46B20, 54C30

DOI:
https://doi.org/10.1090/S0002-9939-05-08267-5

Published electronically:
June 28, 2005

MathSciNet review:
2163583

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the following extended version of Simons' inequality and present its applications. Let be a set and be a subset of . Let be a subset of a Hausdorff topological vector space which is invariant under infinite convex combinations. Let be a bounded function such that the functions are convex for all and whenever , and Let be a sequence in . Assume that, for every , there exists satisfying . Then

If , then the set in the above inequality can be replaced by .

**[AG]**María D. Acosta and Manuel Ruiz Galán,*New characterizations of the reflexivity in terms of the set of norm attaining functionals*, Canad. Math. Bull.**41**(1998), no. 3, 279–289. MR**1637649**, https://doi.org/10.4153/CMB-1998-040-x**[DF]**Robert Deville and Catherine Finet,*An extension of Simons’ inequality and applications*, Rev. Mat. Complut.**14**(2001), no. 1, 95–104. MR**1851724**, https://doi.org/10.5209/rev_REMA.2001.v14.n1.17044**[FG]**M. FABIAN, G. GODEFROY,*The dual of every Asplund space admits a projectional resolution of the identity,*

Studia Math.**91**(1988), 141-151. MR**0985081 (90b:46032)****[FHHMPZ]**Marián Fabian, Petr Habala, Petr Hájek, Vicente Montesinos Santalucía, Jan Pelant, and Václav Zizler,*Functional analysis and infinite-dimensional geometry*, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 8, Springer-Verlag, New York, 2001. MR**1831176****[GS]**M. Ruiz Galán and S. Simons,*A new minimax theorem and a perturbed James’s theorem*, Bull. Austral. Math. Soc.**66**(2002), no. 1, 43–56. MR**1922606**, https://doi.org/10.1017/S0004972700020669**[G1]**G. GODEFROY,*Boundaries of a convex set and interpolation sets,*

Math. Ann.**277**(1987), 173-184. MR**0886417 (88f:46037)****[G2]**Gilles Godefroy,*Some applications of Simons’ inequality*, Serdica Math. J.**26**(2000), no. 1, 59–78. MR**1767034****[GZ]**Gilles Godefroy and Václav Zizler,*Roughness properties of norms on non-Asplund spaces*, Michigan Math. J.**38**(1991), no. 3, 461–466. MR**1116501**, https://doi.org/10.1307/mmj/1029004394**[HHZ]**P. HABALA, P. H´AJEK, V. ZIZLER,*Introduction to Banach Spaces*, I,

Charles University, Prague, 1996.**[L]**Åsvald Lima,*Property (𝑤𝑀*) and the unconditional metric compact approximation property*, Studia Math.**113**(1995), no. 3, 249–263. MR**1330210**, https://doi.org/10.4064/sm-113-3-249-263**[O1]**Eve Oja,*A proof of the Simons inequality*, Acta Comment. Univ. Tartu. Math.**2**(1998), 27–28. MR**1714730****[O2]**Eve Oja,*Géométrie des espaces de Banach ayant des approximations de l’identité contractantes*, C. R. Acad. Sci. Paris Sér. I Math.**328**(1999), no. 12, 1167–1170 (French, with English and French summaries). MR**1701379**, https://doi.org/10.1016/S0764-4442(99)80433-9**[O3]**Eve Oja,*Geometry of Banach spaces having shrinking approximations of the identity*, Trans. Amer. Math. Soc.**352**(2000), no. 6, 2801–2823. MR**1675226**, https://doi.org/10.1090/S0002-9947-00-02521-6**[S1]**S. Simons,*A convergence theorem with boundary*, Pacific J. Math.**40**(1972), 703–708. MR**0312193****[S2]**Stephen Simons,*An eigenvector proof of Fatou’s lemma for continuous functions*, Math. Intelligencer**17**(1995), no. 3, 67–70. MR**1347898**, https://doi.org/10.1007/BF03024373

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
39B62,
46A55,
46B20,
54C30

Retrieve articles in all journals with MSC (2000): 39B62, 46A55, 46B20, 54C30

Additional Information

**Kersti Kivisoo**

Affiliation:
Faculty of Mathematics and Computer Science, Tartu University, J. Liivi 2, EE-50409 Tartu, Estonia

Email:
kersti.kivisoo@mail.ee

**Eve Oja**

Affiliation:
Faculty of Mathematics and Computer Science, Tartu University, J. Liivi 2, EE-50409 Tartu, Estonia

Email:
eveoja@math.ut.ee

DOI:
https://doi.org/10.1090/S0002-9939-05-08267-5

Keywords:
Simons' inequality,
convex sets in topological vector spaces,
convex functions,
uniformly convergent convex combinations,
Banach space geometry.

Received by editor(s):
July 2, 2004

Published electronically:
June 28, 2005

Additional Notes:
This research was partially supported by Estonian Science Foundation Grant 5704

Communicated by:
Jonathan M. Borwein

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.