On the Stretch Factor of Polygonal Chains

Let $P=(p_1, p_2, \dots, p_n)$ be a polygonal chain in $\mathbb{R}^d$. The stretch factor of $P$ is the ratio between total length and distance its endpoints, $\sum_{i = 1}^{n-1} |p_i p_{i+1}|/|p_1 p_n|$. For parameter $c \geq 1$, we call $c$-chain if $|p_ip_j|+|p_jp_k| \leq c|p_ip_k|$, for every triple $(i,j,k)$, $1 i 0$, there noncrossing has $\Omega(n^{1/2-\varepsilon})$, sufficiently large constant $c=c(\varepsilon)$; (ii) $O\left(n^{1/2}\right)$, $c\geq regardless whether crossing or noncrossing; (iii) give randomized algorithm can determine, $\mathbb{R}^2$ with $n$ vertices, minimum 1$ which $O\left(n^{2.5}\ \mathrm{polylog}\ n\right)$ expected time $O(n\log n)$ space. These results generalize dimension $d\geq 2$ $\varepsilon>0$, construct $\Omega\left(n^{(1-\varepsilon)(d-1)/d}\right)$; any $O\left((n-1)^{(d-1)/d}\right)$; $c>1$, test an $n$-vertex $\mathbb{R}^d$ $O\left(n^{3-1/d}\