erdos_1043

**Erdős Problem 1043**: 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 $2$? Pommerenke [Po61] proved that the answer is no. [Po61] Pommerenke, Ch., _On metric properties of complex polynomials._ Michigan Math. J. (1961), 97-115.

theorem erdos_1043 :
    answer(False) ↔ ∀ (f : ℂ[X]), f.Monic → f.degree ≥ 1 →
      ∃ (u : ℂ), ‖u‖ = 1 ∧
      volume ((ℝ ∙ u).orthogonalProjection '' levelSet f) ≤ 2

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