Geometric dilation of closed planar curves: New lower bounds

Abstract

Given two points on a closed planar curve, C, we can divide the length of a shortest connecting path in C by their Euclidean distance. The supremum of these ratios, taken over all pairs of points on the curve, is called the geometric dilation of C. We provide lower bounds for the dilation of closed curves in terms of their geometric properties, and prove that the circle is the only closed curve achieving a dilation of π/2, which is the smallest dilation possible. Our main tool is a new geometric transformation technique based on the perimeter halving pairs of C.

Quadrisecants give new lower bounds for the ropelength of a knot

“…We note that the special case of this formula when r = s also appears in recent papers by Dumitrescu, Ebbers-Baumann, Grüne, Klein and Rote [9,10] investigating the geometric dilation (or distortion) of planar graphs. Gromov had given a lower bound for the distortion of a closed curve (see the paper by Kusner and Sullivan [14]); in [9,10] sharper bounds in terms of the diameter and width of the curve are derived using this minimum-length arc avoiding a ball. (Although the bounds are stated there only for plane curves they apply equally well to space curves.)…”

On the Stretch Factor of Convex Polyhedra whose Vertices are (Almost) on a Sphere

“…Ebbers-Baumann et al [5] have proved that, for a convex cycle C, Dil (C) is well-defined. That is, the maximum in the definition of Dil (C) exists.…”

On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces

“…In the Euclidean case, Zindler curves are not necessarily circles, have many interesting characterizations and are strongly related to other concepts, such as curves of constant area-halving distance and curves of constant width (see Section 2 of the survey [75]). Zindler curves are also related to the construction of graphs of low geometric dilation; see [43,44,48] for investigations of the geometric dilation problem in the Euclidean plane, and [79] for the extension of this problem to Minkowski planes.…”