Denote $D=\{x^2+y^2\le1\}\subset\mathbb R^2$ a disk.


Let $f:D\to\mathbb R$ be a continuous function on it. I am interested in restrictions of simple Morse functions on $\mathbb R^2$, but I suspect the answer is the same for any continuous or any smooth function.


A contour of $f$ is a **connected component** of the preimage of a point, $f^{-1}(\mathrm{const})$.


The diameter of a set is the maximum distance between its points.


Can the maximum diameter of a contour be made arbitrarily small by a suitable choice of $f$?


I suspect that no; namely, I suspect that at least one contour of $f$ will have diameter greater than, say, the radius of the disk, but I can't prove it.


The diameter of the disk is not the bound: Consider a Y-shaped figure formed by three radii, and $f$ being the distance from this figure. Then the largest contour is the Y figure, with the diameter greater than the radius of the disk but smaller than the diameter of the disk.



Can all contours of a function on a disk be made arbitrarily small?
Every continuous function on a unit disk $D^2$ has a level set containing a connected component of diameter at least $\sqrt{3}$; this constant cannot be increased. Generally, in case of $D^n$, $n>2$, there is a level set with a connected component of diameter at least $2$.


See the paper "Level Sets on Disks" by A. Maliszewski and M. Szyszkowski, 2014, <https://doi.org/10.4169/amer.math.monthly.121.03.222>


