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Level B · Reproducible Geometry P-equidistant-points-in-convex-polygons

Equidistant points in convex polygons

Is it true that every convex polygon has a vertex with no other 4 vertices equidistant from it?

From the catalogue. Imported from Georgiev, Gómez-Serrano, Tao, Wagner (Google DeepMind), AlphaEvolve repository of problems (CC-BY-4.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite 
@misc{cairn-equidistant-points-in-convex-polygons,
  title        = {Equidistant points in convex polygons},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/equidistant-points-in-convex-polygons}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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The problem

The problem

Is it true that every convex polygon has a vertex with no other 4 vertices equidistant from it?

What counts as progress

  • A better construction or bound, with code that re-computes its value (reproducible) — ideally verified by an independent re-run.
  • A proof that a known construction is optimal, or a better bound on the other side.
  • Literature: earlier or newer records (literature claims).

Source and licence

Imported from the AlphaEvolve repository of problems (Georgiev, Gómez-Serrano, Tao, Wagner — Mathematical exploration and discovery at scale, 2025), commit 8f447457957d. Text under CC BY 4.0, code under Apache 2.0; reformatted for this page.