I am currently trying to find a way to define some notion of "roundness" for subtours in graphs and that definition should only be based on the comparing (sums of) edge length and on the order in which vertices are encountered along (closed) paths; this is however not my question.

One idea was to investigate the unimodality of the radius function of circles in polar coordinates in the special case that the origin is on the circle.

The next thing to do was to check, how flat an unimodal ellipse could be; the outcome was, that exactly the ellipses with $\frac{a}{b} \ge \frac{1}{\sqrt{2}} \ \wedge\ a\le b$ are the unimodal ones (when $a$ and $b$ are the minor, resp. major ellipse-radius).

Is $\frac{1}{\sqrt{2}}\ $the smallest ratio $\frac{a}{b}$ of smallest width over largest width, that an unimodal curve can have and, if no, what is the lower bound?

Remark: unimodal ellipses can also be charactarized by the fact that their evolute is entirely contained in the closed elliptical disk, the critical case being that two of the astroide's cusps are on the boundary.
It seems however to be unwritten law not to depict evolutes that do not penetrate the boundary.


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