The Dvoretzky-Hanani lemma for rectangles (Q1915634)

For symmetric convex bodies \(U,V\) in the plane, \(\sigma (U,V)\) is the smallest positive number \(r\) such that for any finite sequence \(u_1, \dots, u_n \in U\) there are signs \(\varepsilon_i\) with \(\varepsilon_1 u_1 + \cdots + \varepsilon_k u_k \in r^V\) for \(k = 1, \dots, n\). Those rectangles \(R\) with sides parallel to the coordinate axes are characterized for which \(\sigma (K,R) \leq 1\), where \(K\) is the unit ball of the maximum norm. It is mentioned that there is a relation to rearrangements of series in locally convex spaces.