On average distances and the geometry of Banach spaces (Q1582121)

For a finite set \(F=\{x_1,\dots{},x_n\}\) in the unit sphere \(S\) of the real Banach space \(X\) let \(\mu(F,\cdot):S\rightarrow\mathbb R_+\) be the function defined by \[ \mu(F,x)=\frac 1{n}\sum_{i=1}^n\|x-x_i\|,\quad x\in S, \] and let \(\mu(F,S)\) be its range. Let \(\mu_2(F):=\sup\{\mu(F,S)\}\) and let \(\mu_2(X):=\inf\{\mu_2(F):F\subseteq S, F\) finite\}. The authors consider also the constant \[ T(X):=\inf\{\sup_{y\in S}\inf_{x\in F} \|x-y\|: F\subseteq S, F \text{ finite}\}, \] attached to \(X.\) Main results: a) The conditions \(T(X)<2\) and \(\mu_2(X)<2\) are equivalent. b) If \(\mu_2(X)=T(X)=2\) then \(X\) contains isomorphically \(\ell_1.\) c) Relations between the reflexivity, uniformly non-squareness of \(X\) and the behaviour of \(\mu_2(X)\) are also proved.