Lebesgue universal covering constant

Upper bound, last-listed row

| $$\le 0.8440935944$$ | [Gib2018] | |tracked since 2026-07-24 — no movement seen yet

Lower bound, last-listed row

| $0.832$ | [BS2005] | Rigorous computer-aided search using a circle, equilateral triangle, and regular pentagon |tracked since 2026-07-24 — no movement seen yet

Checked against material for its cited source

• Another row, not one shown above: bound row $0.8271$ line 25 citing [Elek1994] — the claim we checked: that the cited 1994 reference gives the lower bound this row lists, 0.8271, using regular 3^j-gons as well, as the row's comment says. What we read supports it: the paper states that applying its results to all regular 3^j-gons gives 0.8271, and its Corollary 5 states that area bound in closed form. source link (read in the journal edition, Discrete & Computational Geometry 12 (1994), pp. 439–449, all eleven pages, from an Internet Archive copy of the publisher's PDF). Selected: drawn by its position in the list of cited rows, not because it looked wrong.
• Another row, not one shown above: bound row $$2-\frac{2}{\sqrt3}=0.8452994616\dots$$ line 13 citing [Pal1920] — the claim we checked: that Pál's 1920 paper gives the bound 2 − 2/√3 ≈ 0.8453 by truncating the hexagon with an inscribed regular dodecagon, as the row's comment says. What we read supports it: the paper gives the area 2 − 2√3/3, which equals 2 − 2/√3, for a cover made by cutting two corners off the regular hexagon along sides of the inscribed regular dodecagon; the result is an octagon, and the word dodecagon does not appear in the pages read. This verdict does not cover the proof that the octagon is a universal cover. source link (read in the cited printed edition, from page images of the scanned volume (Kgl. Danske Vid. Selsk. Mat.-fys. Medd. III, 2, 1920), pp. 1–24). Selected: drawn by its position in the list of cited rows, not because it looked wrong.

2 bound row(s) here have been checked against material for their cited source (2 drawn by position, 0 chosen because something already looked wrong). Across this whole ledger 70 row(s) have been drawn by position, against 691 rows that name a source, counted on 2026-09-27: 64 were checked against material for their cited sources and for 6 the cited source could not be read at all. 5 more were chosen for suspicion and checked; they are counted apart, because a set selected for suspicion carries no rate. A row not listed here has NOT been checked.