Revisiting the Rectangular Constant in Banach Spaces

Abstract

Let 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 $\mu _p(l_p)$ .

Chasing Convex Bodies and Functions with Black-Box Advice

“…We begin by introducing the rectangular constant µ(X ) of a normed vector space X , which is bounded between √ 2 and 3, with µ(X ) = √ 2 when X is Hilbert and µ( p ) < 3 for any p ∈ (1, ∞) (Joly (1969); Baronti et al (2021)). Next, we define the radial retraction.…”

“…Definition 19 ( (Desbiens, 1990, Definition 2); original from (Joly, 1969, Definition 2)) The rectangular constant µ(X ) of a real normed vector space X is defined as Joly, 1969, Section II), and these bounds are tight: µ(X ) = √ 2 for any Hilbert space (Joly, 1969, Example 1; Section III), and µ(X ) = 3 for "nonuniformly nonsquare" spaces such as 1 and ∞ (Baronti et al (2021)). Moreover, µ( p ) < 3 for all p ∈ (1, ∞).…”