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by "antipodal visibility" of planar, simple polygons I mean the following property:
if two points $p$ and $q$ on the polygon's boundary divide the polygon's boundary into two polylines of equal length, then the line-segment between $p$ and $q$ doesn't intersect the polygon's boundary, except in $p$ and in $q$.

Questions:

  • do non-convex polygons with antipodal visibility exist?
  • if yes, is anything known about their algorithmic construction (esp. of random instances)?
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    $\begingroup$ Take a triangle $ABC$ with $\left|AB\right| = \left|AC\right|$ and with $\measuredangle BAC$ very small (thus an isosceles triangle with an apex very high up). Let $H$ be its orthocenter. I am fairly sure $ABHC$ has antipodal visibility (and is not convex). $\endgroup$ Oct 8, 2014 at 6:31
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    $\begingroup$ Any star polygon symmetrical wrt its center is antipodally visible. $\endgroup$ Oct 8, 2014 at 6:49
  • $\begingroup$ @darijgrinberg I agree; your construction seems to prove existence. $\endgroup$ Oct 8, 2014 at 6:52
  • $\begingroup$ @IlyaBogdanov very nice and obviously true. $\endgroup$ Oct 8, 2014 at 8:36

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