DISPROVED
This has been solved in the negative.
Let $f\in \mathbb{C}[x]$ be a monic polynomial. Must there exist a straight line $\ell$ such that the projection of\[\{ z: \lvert f(z)\rvert\leq 1\}\]onto $\ell$ has measure at most $2$?
A problem of Erdős, Herzog, and Piranian
[EHP58].
Pommerenke
[Po61] (using his previous work
[Po59]) proved that the answer is no, and there exists such an $f$ for which the projection of this set onto every line has measure at least $2.386$. On the other hand, Pommerenke also proved there always exists a line such that the projection has measure at most $3.3$.
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This page was last edited 13 October 2025.
Additional thanks to: msellke
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #1043, https://www.erdosproblems.com/1043, accessed 2025-11-16
