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<p>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, these problems were investigated by Goodman, Johansen, Ramsey, and Frydman among others, in connection with the embeddability problem for non-homogeneous Markov processes.</p>

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Decreasing Paths of Polygons

  • 期刊论文

  • 发表日期：

    2025年6月26日

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  • Isaac Kulp,
  • Charlotte Ochanine,
  • Logan Richard,
  • Leonel Robert,
  • Scott Whitman

摘要

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, these problems were investigated by Goodman, Johansen, Ramsey, and Frydman among others, in connection with the embeddability problem for non-homogeneous Markov processes.