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Let $A$ be a connected compact domain with smooth boundary in the Euclidean 2-plane. Assume its diameter is at most $d$. Assume that the second fundamental form of the boundary is at most $-c$ where $c\geq 0$ (equivalently, the curvature of the boundary is at most $-c$).

Is there an upper estimate on the length of the boundary in terms of $d,c$?

Remark. If $c=0$ then the domain $A$ is convex and hence the length of the boundary is known to be at most $2\pi d$.

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No, there is no bound --- yinyang is our friend.

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This example works if $d\cdot c>2$; otherwise there should be an upper bound.

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  • $\begingroup$ It looks like the diameter of this thing will be large (especially for $|c|$ small. No? $\endgroup$
    – Igor Rivin
    Commented Aug 26, 2017 at 20:33
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    $\begingroup$ @IgorRivin: I don't think so: the curvature constraints is not felt where it matters, in the spiral, where you can do arbitrary many turns close together. $\endgroup$
    – Benoît Kloeckner
    Commented Aug 26, 2017 at 20:59

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