On Ulam's Segment Motion Problem

We study extremal rigid motions of a unit segment in $\mathbb{R}^d$, $d\ge 2$. Given two prescribed positions of a unit segment, we consider continuous motions transforming the initial position into the final one and investigate the total length of the trajectories traced by its endpoints. This minimization problem was posed by Ulam~\cite{Ulam1960} and solved by Gurevich~\cite{Gurevich1977} and Dubovitskii~\cite{Dubovitskii1976}. Two natural lower bounds are given by the sum of the endpoint displacements and by the angle between the initial and final directions of the segment. We characterize all pairs of segment positions for which either of these lower bounds is attained. In arbitrary dimension, we obtain complete characterizations of the equality cases for both the endpoint-displacement bound and the angular bound.