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Jul 29, 2017 at 12:11 history edited Joseph O'Rourke CC BY-SA 3.0
Image link broken; now fixed.
Apr 7, 2013 at 19:43 comment added Yoav Kallus It seems from numerics that it is indeed the case that $f$ is weakly unimodal. So this is a valuable example to keep in mind. However, as I have pointed out, I am interested in the cases where $f(\theta)$ is not constant on any interval. Particularly, I want to eliminate the cases that $K$ or $L$ are circular disks and the cases where $R_\theta(L)\subset K$ for some $\theta$.
Apr 7, 2013 at 19:19 comment added Joseph O'Rourke @Sergei: I took the liberty of adding a figure. Hope I illustrated your pretty idea correctly.
Apr 7, 2013 at 19:18 history edited Joseph O'Rourke CC BY-SA 3.0
added 276 characters in body; deleted 6 characters in body
Apr 7, 2013 at 18:44 history answered Sergei Ivanov CC BY-SA 3.0