Convexity and Haar null sets

Author:
Eva Matousková

Journal:
Proc. Amer. Math. Soc. **125** (1997), 1793-1799

MSC (1991):
Primary 46B10; Secondary 46B20

DOI:
https://doi.org/10.1090/S0002-9939-97-03776-3

MathSciNet review:
1372040

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Abstract: It is shown that for every closed, convex and nowhere dense subset of a superreflexive Banach space there exists a Radon probability measure on so that for all . In particular, closed, convex, nowhere dense sets in separable superreflexive Banach spaces are Haar null. This is unlike the situation in separable nonreflexive Banach spaces, where there always exists a closed convex nowhere dense set which is not Haar null.

**[A]**N. Aronszajn,*Differentiability of Lipschitzian mappings between Banach spaces*, Studia Math. LVII (1976), 147-190. MR**54:13562****[AL]**D. Amir and J. Lindenstrauss,*The structure of weakly compact sets in Banach spaces*, Ann. of Math. 88 (1968), 35-46. MR**37:4562****[BP]**C. Bessaga and A. Pelczynski,*Selected topics in infinite-dimensional topology*, PWN, Warszawa 1975. MR**57:17657****[BN]**J.M. Borwein and D. Noll,*Second order differentiability of convex functions in Banach spaces*, Trans. A.M.S. 342 (1994), 43-81. MR**94e:46076****[C]**J.P.R. Christensen,*Topology and Borel structure*, North-Holland, Amsterdam 1974. MR**50:1221****[GG]**V.I. Gurarii and N.I. Gurarri,*On basis in uniformly convex and uniformly smooth spaces*, Izv. Akad. Nauk SSSR 35 (1971), 210-215. MR**44:780****[J]**R.C. James,*Super-reflexive spaces with bases*, Pacific J. of Math., Vol. 41, 2 (1972), 409-419 MR**46:7866****[LT]**J. Lindenstarauss and L. Tzafriri,*Classical Banach spaces I.*, Springer-Verlag, Berlin 1977. MR**58:17766****[MS]**E. Matousková and C. Stegall,*A characterization of reflexive Banach spaces*, to appear in Proc. A.M.S. MR**96g:46005****[MZ]**E. Matousková and L. Zajícek,*Second order differentiability of integral functionals*, to appear in Czechoslov. Math. Journ.**[PT]**D. Preiss and J. Tiser,*Two unexpected examples concerning differentiability of Lipschitz functions on Banach spaces*, to appear in Proceedings of GAFA seminar, Tel Aviv.**[S]**C. Stegall,*A proof of the theorem of Amir and Lindenstrauss*, Israel Jour. of Math., 68 (1989), 185-192. MR**91b:46017****[V]**L. Vasák,*On one generalization of weakly compactly generated Banach spaces*, Studia Math. 70 (1981), 11-19. MR**83h:46028**

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Additional Information

**Eva Matousková**

Affiliation:
Department of Mathematical Analysis, Charles University, Sokolovská 83 , 18600 Prague, Czech Republic;
Institut für Mathematik, Johannes Kepler Universität, Altenbergerstraße, A-4040 Linz, Austria

Email:
eva@caddo.bayou.uni-linz.ac.at

DOI:
https://doi.org/10.1090/S0002-9939-97-03776-3

Keywords:
Superreflexive Banach spaces,
convexity,
Haar null sets

Received by editor(s):
February 22, 1995

Received by editor(s) in revised form:
January 8, 1996

Additional Notes:
The author was partially supported by the grant GAČR 201/94/0069 and by a grant of the Austrian Ministry of Education.

Communicated by:
Dale Alspach

Article copyright:
© Copyright 1997
American Mathematical Society