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Dec 10 at 13:30 comment added Beni Bogosel @guest61: the estimate you want holds in the other sense, as indicated in my answer.
May 24, 2022 at 14:23 comment added guest61 Many thanks for the detailed answer and the counter example. Unfortunately I need the constant $1$ and the power $2$. For the disk the desired estimate holds with equality. Maybe there is some hope that it holds for a regular $N$ -polygon for $N$ large enough. Otherwise the reverse inequality may have a chance for all $N$.
May 24, 2022 at 12:32 history edited Denis Serre CC BY-SA 4.0
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May 24, 2022 at 12:13 history answered Denis Serre CC BY-SA 4.0