Let $f\in \mathbb{C}[x]$ be a monic non-constant polynomial. Must there exist a straight line $\ell$ such that the projection of$$\{z:|f(z)|\le 1\}$$onto $\ell$ has measure at most $2$?

AMS 28 30 · solved (literature)

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