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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.

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    $\begingroup$ At a minimum you need $f$ to be continuous or something. Without any other assumptions you can choose an $f$ which is injective, and then every contour is a single point and has diameter zero. $\endgroup$
    – Nate Eldredge
    Commented Apr 22, 2018 at 20:39
  • $\begingroup$ @NateEldredge Right, will edit the question. $\endgroup$
    – Alexander Gelbukh
    Commented Apr 22, 2018 at 20:45
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    $\begingroup$ For a simple Morse function, consider the contour at the level $f(0)$ passing through $0$. If it reaches the boundary, you are done. Otherwise it bounds some domain. Move the level so that this domain is expanding and consider the first moment it reaches the boundary (possibly by merging with another domain). Shortly before or shortly after that moment you should have what you want. $\endgroup$
    – fedja
    Commented Apr 23, 2018 at 1:41

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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

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