On a problem of H. Freudenthal (Q1388213)

In 1963, Freudenthal posed the following problem: When is the inequality \[ \begin{aligned} \sum_{i=1}^n|a_i|-\sum_{1\leq i<j\leq n}|a_i+a_j|+&\sum_{1\leq i<j<k\leq n}|a_i+a_j + a_k|- \cdots\\ &+(-1)^{n-1}|a_1+a_2+ \ldots+a_n|\geq 0 \end{aligned}\tag{1} \] always true for all \(a_1, a_2, \cdots,a_n \in \mathbb R^m ?\) Obviously, (1) is true if \(n=1,2.\) If \(n=3\) , (1) is the Hlawka inequality \[ |a|+|b|+|c|-|a+b|-|b+c|-|b+c|+|a+b+c|\geq 0. \] But if \(n=4\), Luxemburg (1970) gave a counter-example showing that (1) is not true, with \(a_i=c (i=1,2,3), a_4=-2c (c\neq 0).\) In the present paper the authors construct counter-examples for all \(n \geq 5\). As a result, it is shown that the inequality (1) is always true for arbitrary \(a_1,a_2,\ldots,a_n \in \mathbb R^m\) if and only if \(n=1,2,3.\)