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Let P be a convex polygon in the plane, and let P’ be an enlarged version of P, dilated by a scale factor of 2. Show that seven copies of P can completely cover P’.

I vaguely remember seeing this problem online, but I can’t find the source.

I noticed that seven circles of radius 1 can cover a circle of radius 2. Their intersections also trace a regular hexagon. Is that useful? I couldn’t make much progress.

Avery Wenger
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  • Some context to add to question : $\mathbb R^2$ is a Doubling space with constant $7$ as mentioned in the post (Kindly add to your post). In any case, you can perhaps try to look for a bijection from $\mathbb R^2$ to itself such that the unit circle centred at the origin maps to a copy of $P$ with $0$ in its interior. Doing this, you can probably map the entire ball covering picture into a polygon covering picture. – Sarvesh Ravichandran Iyer Feb 28 '24 at 09:27
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    A solution to this problem can be found here – Anay Gautam Feb 29 '24 at 07:40
  • @AnayGautam There is no solution on that page. – Avery Wenger Mar 06 '24 at 02:19

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