On a Question of Erdős on Lemniscates in R^m: Connected Lemniscates Lie in a Ball of Radius 2

For points x_1, …, x_n in R^m let E = {x ∈ R^m : ∏_{i=1}^n |x − x_i| ≤ 1}. A question of Erdős, posed in Hayman's collection of problems of 1974 and recorded as Problem 4.23 by Hayman and Lingham, asks how large the volume of E can be and whether a connected E can be covered by a ball of radius 2; for m = 2 the covering statement is a theorem of Pommerenke (1959). We answer the covering question: if E is connected, then |x − c| < 2 for every x ∈ E, where c is the centroid of x_1, …, x_n, for all m and n. Projecting E onto the line through c and a point of E reduces the question to Pommerenke's planar theorem for a polynomial with real zeros; we prove that theorem again, in the closed-set form needed. Collinear Chebyshev configurations give connected sets E with max_{x∈E} |x − c| = 2·2^(−1/n), so the constant 2 cannot be lowered uniformly in n, already for m = 1. The same projection shows that for each n the best radius about the centroid is the same in every dimension; it equals 2·2^(−1/n) for n ≤ 2, and whether it does for all n, as numerical experiments suggest, is open, as is the volume question. The question is Erdős's and the planar case is Pommerenke's; the note contributes the reduction, which gives the covering statement in every dimension and the dimension-independence of the best radius about the centroid. We found no proof of the covering statement for m ≥ 3 in the sources we searched. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: AMR-022-4023 (Hayman–Lingham, Research Problems in Function Theory, Problem 4.23 (P. Erdős); arXiv:1809.07200v2).