Does the diagonal bound property imply that X is a Hilbert space? (Q1263040)

Let X be a Banach space and let K be a non-negative real number. If, for all \(x_ 1,...,x_ n\in X\) the inequality \[ (1/K)\inf_{\epsilon (n)}\| \sum^{n}_{i=1}\epsilon_ ix_ i\|^ 2\leq \sum^{n}_{i=1}\| x\quad_ i\|^ 2\leq K\sup_{\epsilon (n)}\| \sum^{n}_{i=1}\epsilon_ ix_ i\|^ 2 \] holds, X is said to have the diagonal bound property (DBP) for the constant K. The authors show that: (i) X is isometrically isomorphic to an inner product space iff X has the DBP for \(K=1,\) (ii) if X has the DBP for \(1\leq K<2\) then X is reflexive, and (iii) if X has the DBP for any K, then X is of type 2-\(\epsilon\) and of cotype \(2+\epsilon\) for each \(\epsilon >0.\) The necessity of X having the DBP for \(K=1\) in (i) is a result of Abreu and Alonso, while the sufficiency has to do with the parallelogram property. Additional results relate properties of Banach lattices to the DBP.