# Isoperimetric inequality on the plane

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 $2\pi d$.

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

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

• It looks like the diameter of this thing will be large (especially for $|c|$ small. No? – Igor Rivin Aug 26 '17 at 20:33
• @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. – Benoît Kloeckner Aug 26 '17 at 20:59