Claims

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1958_12_01_erdos_herzog_piranian: A line segment or a closed disc of transfinite diameter at least one has minimal lemniscate area zero; the problem's source derives the two cases from Chebyshev polynomials and from zeros on a circle.

2024_02_03_krishnapur_lundberg_ramachandran: A compact set of capacity above one whose equilibrium measure obeys a power bound on discs has minimal lemniscate area zero (Corollary 1.6 of the journal version of Inradius of random lemniscates).

2025_03_24_krishnapur_lundberg_ramachandran: For the closure K of a bounded open set with C^2 boundary and capacity one, the minimal area of {|p| < 1} over monic polynomials with zeros in K has infimum zero over the degrees; preprint.

2026_01_29_feng: Two countable compact sets of transfinite diameter zero, one with minimal lemniscate area at least pi/4 and one with minimal area below any prescribed bound, so the minimal area is not a function of the transfinite diameter.

2026_04_03_ghosh_ramachandran: Compact sets of equal capacity in (0, 1) with different minimal areas, so the minimal area is not determined by the capacity; also, for compact K of capacity t > 1, the degree-n minimal area of {|p| < 1} decays like t^(-2n).

2026_09_05_tzachristas: Every compact set of capacity one admits monic polynomials with zeros in it and unit lemniscates of arbitrarily small area; hence every closed infinite set of transfinite diameter at least one has minimal area zero.

2026_09_06_gessel: For every closed infinite set of transfinite diameter at least one, the areas of {|p| < 1} over monic polynomials with all roots in the set have infimum zero; Lean checked by an outside verifier, not built here.

2026_09_06_kitamura: An independent Lean 4 development proving that a closed infinite set whose infimum of finite Fekete diameters is at least one has minimal lemniscate area zero; the bridge to the classical transfinite diameter is not formalized.

2026_09_06_shlummi: Every closed infinite set of transfinite diameter at least one admits monic polynomials with all roots in it and unit lemniscates of arbitrarily small area; author-run Lean, not built here.

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