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Staircase Connected Sets

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Abstract

A compact set \(S \subset {\Bbb R}^2\) is staircase connected if every two points \(a,b \in S\) can be connected by a polygonal path with sides parallel to the coordinate axes, which is both x-monotone and y-monotone. \(\xi(a,b)\) denotes the smallest number of edges of such a path. \(\xi(\cdot,\cdot)\) is an integer-valued metric on S. We investigate this metric and introduce stars and kernels. Our main result is that the r-th kernel is nonempty, compact and staircase connected provided \(r \ge \frac{1}{2} \cdot {\it stdiam}(S) +1\).

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Correspondence to Evelyn Magazanik or Micha A. Perles.

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Magazanik, E., Perles, M. Staircase Connected Sets. Discrete Comput Geom 37, 587–599 (2007). https://doi.org/10.1007/s00454-007-1308-9

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Keywords

  • Discrete Comput Geom
  • Vertical Edge
  • Vertical Segment
  • Horizontal Segment
  • Horizontal Edge