Abstract
AbstractLet X be a real Banach space. The rectangular constant $\mu (X)$ and some generalisations of it, $\mu _p(X)$ for $p \geq 1$ , were introduced by Gastinel and Joly around half a century ago. In this paper we make precise some characterisations of inner product spaces by using $\mu _p(X)$ , correcting some statements appearing in the literature, and extend to $\mu _p(X)$ some characterisations of uniformly nonsquare spaces, known only for $\mu (X)$ . We also give a characterisation of two-dimensional spaces with hexagonal norms. Finally, we indicate some new upper estimates concerning $\mu (l_p)$ and $\mu _p(l_p)$ .
Highlights
The vector x is (Birkhoff–James) orthogonal to y if x ≤ x + λy for every real λ
In [4], the equivalence was extended to two-dimensional spaces
We recall that a space X is nonuniformly nonsquare if for every ǫ > 0 there exist x, y ∈ S(X) such that x ± y > 2 − ǫ
Summary
In [8], Joly proved that 2 ≤ μ(X) ≤ 3 and, for dim(X) ≥ 3, that μ(X) = 2 if and only if X is a Hilbert space. In [1], the following result was proved: μ(X) = 3 if and only if the space X is nonuniformly nonsquare. In [6], Gastinel and Joly extended the definition of the rectangular constant: for p ≥ 1, x, y ∈ S(X) and x ⊥ y, μp(x, y) = sup λ≥0. In Theorem 2.1 we will revise this result by proving that, for p ≥ 2, X is a Hilbert space if and only if μp(X) = 1.
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