Consider $n$ points $x_1,\ldots,x_n$ on the unit circle $S^1\subset \mathbb{C}$. Suppose these are ordered such that $\sum_{i=1}^n |x_i-x_{i+1}|$ (indices modulo $n$) is maximal for all possible permutation of indices, i.e. going from $x_1$ to $x_2$ to $x_3$ ... to $x_n$ to $x_1$ on straight lines in the complex plane describes the longest possible path to visit each $x_i$ exactly once. Let $c_i:=|x_i-x_{i+1}|$ for $i=1\ldots n$ and $\sigma$ a permutation such that $$c_{\sigma(1)}\geq c_{\sigma(2)}\geq \ldots \geq c_{\sigma(n)}.$$

For an arbitrary point $z\in S^1$ define $d_i:=d_{z,i}:=|z-x_i|$ and let $\tau:=\tau_z$ be a permutation such that $$d_{\tau(1)}\geq d_{\tau(2)}\geq \ldots \geq d_{\tau(n)}.$$

The question is: ** Is there a choice for $z$ such that for all $i=1\ldots n$ the inequalities $d_{\tau(i)}\leq c_{\sigma(i)}$ hold? **