Positivity 8: 269–281, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Almost ﬂat locally ﬁnite coverings of the sphere
and C. ZANCO
Department of Mathematics, Ben-Gurion University of the Negev P.O. Box 653, Beer-Sheva
84105, Israel. E-mail: email@example.com
Dipartimento di Matematica “F. Enriques”, Università degli Studi, Via C. Saldini, 50, 20133
Milano MI, Italy. E-mail: firstname.lastname@example.org
(Received 7 November 2001; accepted 3 December 2001)
Abstract. For any subset A of the unit sphere of a Banach space X and for ∈02 the notion
of -ﬂatness is introduced as a “measure of non-ﬂatness” of A. For any positive , construction of
locally ﬁnite tilings of the unit sphere by -ﬂat sets is carried out under suitable -renormings of X
in a quite general context; moreover, a characterization of spaces having separable dual is provided
in terms of the existence of such tilings. Finally, relationships between the possibility of getting such
tilings of the unit sphere in the given norm and smoothness properties of the norm are discussed.
Math. Subject Classiﬁcation 1991: 46B10, 46B20.
Key words: -ﬂat set, locally ﬁnite covering, duality mapping.
The present paper is concerned with coverings and tilings of spheres in Banach
A family of subsets of a topological space Y is a covering of Y if each element
of Y belongs to at least one member of ; is locally ﬁnite provided each point of
Y has a neighborhood that meets just ﬁnitely many members of . As the term is
used here, a proper subset of Y is a body whenever it is the closure of its nonempty
interior (no assumption of connectedness is made here). A tiling of Y is a covering
of Y by bodies (tiles) such that each element of Y is interior to at most one member
of the covering.
In the context of inﬁnite-dimensional Banach spaces, ﬁnding locally ﬁnite cov-
erings may be really difﬁcult or even impossible as soon as we ask that the members
of the covering satisfy “nice” properties. For instance, no Banach space contain-
ing an inﬁnite-dimensional reﬂexive subspace can be locally ﬁnitely covered by
bounded convex subsets . Also a separable Banach space can be locally ﬁnitely
tiled by bounded convex tiles if and only if it is isomorphic to a polyhedral space
. In any separable polyhedral space X such a tiling can be obtained in a natural
way by considering the true faces of the unit ball B
under some suitable renorming
(see Section 1 for deﬁnitions) and intersecting the positive closed cones generated
by such faces (having the origin as their apex) with the closure of the “spherical
, n∈N (of course, B
is a tile too; see  for details).