A distribution maximum inequality for rearrangements of summands (Q2904079)

Let \(x_1,\dotsc,x_n\) be elements of a normed space \((X,\left\| \cdot\right\|\)) with \(\sum_{i=1}^nx_i=0\), and let \(\Pi_n\) be the set of all permutations of \((1,\dotsc,n)\). In this paper, it is shown that there exists a constant \(C>0\) such that NEWLINE\[NEWLINE \left |\left\{ \pi\in\Pi_n:\,\max_{1\leq k\leq n} \left\| \sum_{i=1}^k x_{\pi(i)}\right\| > t\right\}\right | \leq C \left |\left\{ \pi\in\Pi_n:\,\max_{1\leq k\leq n} \left\| \sum_{i=1}^k \vartheta_i x_{\pi(i)}\right\| > t/C\right\}\right | NEWLINE\]NEWLINE for any collection \(\vartheta_1,\dotsc,\vartheta_n\in\{-1,1\}\) of signs and any \(t>0\), where \(|A|\) denotes the number of elements of a finite set \(A\).