Estimates for Kottman's separation constant in reflexive Banach spaces

Abstract

Abstract. In reflexive Banach spaces with some degree of uniform convexity, we obtain estimates for Kottman's separation constant in terms of the corresponding modulus.

Constructing separated sequences in Banach spaces

“…This covers the case when X is superreflexive. We find the values of s(L p ) and show that our estimates for s(X) are sharper than some of the estimates obtained in [6] and [25].…”

“…In contrast to the constants considered in [29], [6] and [25], one of our lower bounds for s(X) is isomorphically invariant. Our results give new answers to Diestel's problem.…”

“…In [25], it was shown that K(X) ≥ g(X). From Theorem 3 we see that s(X) ≥ λ(X) = λ(Y ) for any space Y isomorphic to X.…”

“…Partial answers to Diestel's problem were obtained in [29], [6] and [25]. The results given in those papers guarantee the existence of sequences with the separation constants bounded from below by some coefficients or values of some moduli, in particular the modulus of convexity.…”

A note on symmetric separation in Banach spaces

“…The above proposition allows us to obtain an interesting estimate for K s (X) in terms of K(X) for certain classes of Banach spaces. In order to explain this, we need some results from [8] and [20]. In the paper [8], Dronka, Olszowy and Rybarska-Rusinek introduced a constant, denoted γ 0 (X), as follows:…”