Arcs with increasing chords in $\mathbf{R}^d$

Abstract

A curve $\gamma$ that connects $s$ and $t$ has the increasing chord property if $|bc| \leq |ad|$ whenever $a,b,c,d$ lie in that order on $\gamma$. For planar curves, the length of such a curve is known to be at most $2\pi/3 \cdot |st|$. Here we examine the question in higher dimensions and from the algorithmic standpoint and show the following: (I) The length of any $s-t$ curve with increasing chords in $\mathbf{R}^d$ is at most $2 \cdot \left( e/2 \cdot (d+4) \right)^{d-1} \cdot |st|$ for every $d \geq 3$. This is the first bound in higher dimensions. (II) Given a polygonal chain $P=(p_1, p_2, \dots, p_n)$ in $\mathbf{R}^d$, where $d \geq 4$, $k =\lfloor d/2 \rfloor$, it can be tested whether it satisfies the increasing chord property in $O\left(n^{2-1/(k+1)} {\rm polylog} (n) \right)$ expected time. This is the first subquadratic algorithm in higher dimensions.