On the optimality of a theorem of Elton on \(\ell_1^n\) subsystems. (Q5951490)

A well-known theorem of \textit{J. Elton} [Trans. Am. Math. Soc. 279, 113--124 (1983; Zbl 0525.46011)] states: For every \(0<\delta<1\) there exist two positive constants \(C= C(\delta)< 1\) and \(c= c(\delta)< 1\) such that for every sequence \((v_i)_i\) of \(n\) vectors in the unit ball of a real Banach space satisfying \(Ave_{\varepsilon_i= \pm 1}\|\sum^n_{i=1} \varepsilon_i v_i\|\geq\delta n\), there exists a set \(A\subset \{1,2,\dots, n\}\) of cardinality \(| A|\geq cn\) such that \(\|\sum_{i\in A}a_iv_i\|\geq C\sum_{i\in A}| a_i|\) for every choice of scalars \(a_i\). Moreover, as \(\delta\nearrow 1\) one may choose either \(C\nearrow 1\) and \(c\nearrow 1/2\) or \(c\nearrow 1\) (then \(C\to 0\)). For the dependence of \(C\) and \(c\) on \(\delta\) when \(\delta\) is small, see \textit{A. Pajor} [``Sous-espaces \(\ell_1\) des espaces de Banach'' (Hermann, Paris) (1985; Zbl 1032.46021)], \textit{M. Talagrand} [Invent. Math. 107, 41--59 (1992; Zbl 0788.46002)], and \textit{S. Mendelson} and \textit{R. Vershynin} [Invent. Math. 152, 37--55 (2003; Zbl 1039.60016)]. In the aforementioned paper, J. Elton described an example due to S. Szarek which shows that if we fix \(c> 1/2\) in the theorem, then we cannot expect that \(C\to 1\) as \(\delta\to 1\). He also noticed that it is unknown what happens if we substitute the average with the minimum in the hypothesis of the theorem. The authors provide here an example showing that even in this case we cannot obtain a better result. Let \((e_i)_{i\leq n}\) denote the canonical basis of \(\mathbb{R}^n\), \(\|\cdot\|_1\) denote the \(\ell_1\)-norm, and, given \(A\subset\{1,2,\dots, n\}\), let \(P_A\) denote the coordinate projection onto \(\mathbb{R}^A\). The authors prove the following Theorem. Let \(0< \alpha< 1/2\) and \(0< \beta< 1\). For every sufficiently large \(n\) there exists a norm \(\|\cdot\|\) on \(\mathbb{R}^n\) with the following properties: (i) if \(A\subset B\subset \{1,2,\dots, n\}\), then \(\| P_A x\|\leq\| P_B x\|\) for every \(x\in \mathbb{R}^n\), (ii) \(\| e_i\|= 1\) for every \(i\) and \(\|\sum^n_{i=1} \varepsilon_i v_i\|\geq n-\beta\) for every choice of \(\varepsilon_i=\pm 1\), (iii) for every \(A\subset\{1,2,\dots, n\}\) with cardinality \(| A|\geq 1+(\alpha+ 1/2)n\) there exists a vector \(x\in \mathbb{R}^n\) such that \(\| P_Ax\|< (1-\gamma)\| P_A x\|_1\), where \(\gamma= \gamma(\alpha,\beta)\) is of order \(\alpha^3\beta\).