On series of signed vectors and their rearrangements (Q2884006)

The author addresses a long-standing open problem about balancing vectors in \(\ell_2^{(n)}\) (i.e., in \({\mathbb R}^n\) equipped with its standard Euclidian norm): Does there exist an absolute constant \(C\) such that for every finite collection \(\{x_1, \dots , x_m\} \subset \ell_2^{(n)}\) there is a collection of \(\alpha_k = \pm 1\), \(k = 1,2, \dots, m\), such that NEWLINE\[NEWLINE \max_{1 \leq j \leq m} \|\sum_{k=1}^j \alpha_k x_k\| \leq C \sqrt{n} \max_k\|x_k\|\;? NEWLINE\]NEWLINE This type of combinatorial-geometric inequalities have close ties with the theory of series in Banach spaces, see [\textit{M. I. Kadets} and \textit{V. M. Kadets}, Series in Banach spaces: conditional and unconditional convergence. Basel: Birkhäuser (1997; Zbl 0876.46009)].NEWLINENEWLINEIn the paper under review, a number of results close to the aforementioned conjecture are demonstrated. For example, the existence of \(\alpha_k = \pm 1\), \(k = 1,2, \dots , m\), with NEWLINE\[NEWLINE \max_{1 \leq j \leq m} \|\sum_{k=1}^j \alpha_k x_k\| \leq (C_1 \sqrt{n} + C_2 \log m) \max_k\|x_k\| NEWLINE\]NEWLINE is proved. It is also shown that there exists an absolute constant \(C\) such that for every finite collection \(\{x_1, \dots , x_m\} \subset \ell_2^{(n)}\) there is a collection of \(\alpha_k = \pm 1\), \(k = 1,2, \dots , m\), and there is a rearrangement \(\pi\) of \(\{1,2, \dots, m\}\) such that NEWLINE\[NEWLINE \max_{1 \leq j \leq m} \|\sum_{k=1}^j \alpha_k x_{\pi(k)}\| \leq C \sqrt{n} \max_k\|x_k\|. NEWLINE\]NEWLINE An analogous problem for vectors in \(\ell_\infty^{(n)}\) is also tackled.