Every continuous function on a unit disk $D^2$ has a level set containing a connected component of diameter at least $\sqrt{3}$.

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

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

Every continuous function on a unit disk $D^2$ has a level set containing a connected component of diameter at least $\sqrt{3}$.

See the paper "Level Sets on Disks" by A. Maliszewski and M. Szyszkowski, 2014, http://www.jstor.org/stable/10.4169/amer.math.monthly.121.03.222

Every continuous function on a unit disk $D^2$ has a level set containing a connected component of diameter at least $\sqrt{3}$.

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