Problem 97 — Visual explainer (undergrad)
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Problem statement
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Does every convex polygon have a vertex with no other $4$ vertices equidistant from it?

Picture
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[Given objects + constraints]  --->  [Count / structure]  --->  [Show bound / existence]

Question: what to look at
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- Restate the problem as: given the constraint, what asymptotic bound or
  structure must follow?
- Use the comments as benchmarks (best known bounds / constructions).

Benchmarks / known results (from comments)
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- Erd\H{o}s originally conjectured this (in ) with no $3$ vertices
  equidistant, but Danzer found a {IMAGE=97-Danzer,convex polygon} on 9 points
  such that every vertex has three vertices equidistant from it (but this
  distance depends on the vertex).
- Danzer's construction is explained in.
- If this fails for $4$, perhaps there is some constant for which it holds?
- In Erd\H{o}s claimed that Danzer proved that this false for every constant -
  in fact, for any $k$ there is a convex polygon such that every vertex has
  $k$ vertices equidistant from it.