Decreasing paths of polygons
Metric Geometry
2025-06-09 v2 Probability
Abstract
We call a continuous path of polygons decreasing if the convex hulls of the polygons form a decreasing family of sets. For an arbitrary polygon of more than three vertices, we characterize the polygons contained in it that can be reached by a decreasing path (attainability problem), and we show that this can be done by a finite application of "pull-in" moves (bang-bang problem). In the case of triangles, this problems was investigated by Goodman, Johansen, Ramsey, and Frydman among others, in connection with the embeddability problem for non-homogeneous Markov processes.
Keywords
Cite
@article{arxiv.2402.12643,
title = {Decreasing paths of polygons},
author = {Isaac Kulp and Charlotte Ochanine and Logan Richard and Leonel Robert and Scott Whitman},
journal= {arXiv preprint arXiv:2402.12643},
year = {2025}
}