Positivity 7: 99–102, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
Banach Spaces Which Are Far from all Lattices
Department of Mathematics, Haifa 32000, Israel (E-mail: firstname.lastname@example.org)
Abstract. We consider n-dimensional real Banach spaces X which are far, in the Banach–Mazur
distance, from all complemented subspaces of all Banach lattices. We show that this is related to the
volume ratio values of X with respect to ellipsoids and to zonoids.
Recall the deﬁnition of the Banach-Mazur distance between two n-dimensional
real Banach spaces X and Y to be d(X,Y) := infTT
, where the inﬁmum
ranges over all onto isomorphisms T : X → Y . It is a well known result, due to F.
John, that if dim(X) = n then d(X,
Deﬁnition. The local unconditional structure constant of a Banach space X is
deﬁned to be lust(X) := infα : X → Lβ : L → X
where the inﬁmum
ranges over all Banach lattices L and all factorizations j = βα of the natural
embedding j : X → X
It is clear that since
is a lattice, lust(X) ≤ d(X,
dim(X) = n.
The ﬁrst examples of ﬁnite-dimensional Banach spaces for which their lust
constants tend to inﬁnity with the dimensions, were provided in  where it was
proved, among other things, that for the n
-dimensional space of operators on
denoted by L(
), lust (L(
n. The existence of n-dimensional sub-
n was later established in  , however
up to this date it is not known how to construct such spaces.
In this lecture we shall strengthen this result by relating the lust constants to
Deﬁnition. The classical volume ratio, somtimes called the ellipsoidal volume ratio
of an n-dimensonal Banach space X is deﬁned to be
) := inf
where the inﬁmum is taken over all ellipsoids D contained in B
, the unit ball of
The notions of volume ratios are very much connected to the study of the local
structure, i.e., the geometry of ﬁnite-dimensional Banach spaces, see [7, 8].
This lecture was delivered in the Positivity conference, in Nijmegen, Holland, on July 18, 2001