Separated sequences in nonreflexive Banach spaces

Abstract

Abstract. We prove that there is c > 1 such that the unit ball of any nonreflexive Banach space contains a c-separated sequence. The supremum of these constants c is estimated from below by 5 √ 4 and from above approximately by 1.71. Given any p > 1, we also construct a nonreflexive space so that if the convex hull of a sequence is sufficiently close to the unit sphere, then its separation constant does not exceed 2

Separated sequences in uniformly convex Banach spaces

“…A celebrated result of Elton and Odell [4] asserts that the unit sphere of every infinitedimensional Banach space X contains a (1 + µ)-separated sequence for some µ > 0. It was subsequently shown by Kryczka and Prus [7] that the unit sphere of every infinitedimensional nonreflexive Banach space contains a 5 √ 4-separated sequence. For uniformly convex spaces X we use Theorem 1.1 to deduce a lower bound for the separation constant in terms of the modulus of convexity δ X : Theorem 1.2.…”

Equilateral sets in infinite dimensional Banach spaces

“…(The answer is positive in case when X contains some ℓ p , 1 ≤ p < ∞ or c 0 ). We note in this connection that by a result of Elton and Odell the unit sphere of every infinite dimensional Banach space contains an infinite (1 + ε)-separated set for some ε > 0 ( [8], see also [6] and [11]).…”

Banach spaces in which large subsets of spheres concentrate

“…Kottman proved in 1975 ( [26]) that the unit sphere of every infinite dimensional Banach space admits an infinite (1+)-separated subset, which was improved in 1981 by Elton and Odell to (1 + ε)-separated for some ε > 0 ( [7]) who also noted that c 0 (ω 1 ) does not admit an uncountable (1+ε)-separated set. The Kottman constant of a Banach space (the supremum over δ > 0 such that there is an infinite δ-separated subset of the unit sphere) turned out to be an important tool used for investigating the geometry of the space (e.g., [3,28,33,1,34]). In fact, it is related to many aspects of Banach spaces like e.g., packing balls, measures of incompactness, fixed points, average distances and infinite dimensional convexity (see e.g., the papers citing [27] or [7]).…”