Kovac 2024 polygons unit area vertices sets infinite

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theorem_1: Kovač and Predojević's theorem that every measurable planar set of infinite Lebesgue measure contains four concyclic points spanning a non-degenerate quadrilateral of area 1, and by rescaling of any prescribed area a > 0.

theorem_2: Kovač and Predojević's theorem that some planar set of infinite Lebesgue measure has every convex polygon with congruent sides and all vertices in it of area strictly less than 1; the set is 4xy < 1 with x > 1, y > 0.

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Vjekoslav Kovač, Bruno Predojević, Polygons of unit area with vertices in sets of infinite planar measure. arXiv:2412.11725 (2024). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2412.11725), every other right reserved. The paper appeared in Canad. Math. Bull. 69 (2026), no. 3, 849-864, DOI 10.4153/S0008439525101537 (arXiv record and Crossref); the copy read for this card is arXiv v2 (10 November 2025), whose pages the results below cite.

Kovač and Predojević settle two of the questions on unit-area polygons with vertices in planar sets of infinite Lebesgue measure that Erdős reported, from his attempts with Mauldin, in the proceedings of the 1983 Oberwolfach conference Measure Theory. Theorem 1 answers the cyclic-quadrilateral question positively: every measurable planar set of infinite measure contains four concyclic points spanning a non-degenerate quadrilateral of area exactly 1, and by rescaling the set, of any prescribed area a > 0. Theorem 2 answers the final question of the series negatively and in strengthened form: some planar set of infinite Lebesgue measure has every convex polygon with congruent sides and all vertices in the set of area strictly less than 1, so the answer is no whether the number of sides n is fixed or may depend on the set. The paper calls its approach to Theorem 1 soft (qualitative, measure-theoretic): pigeonholing over quadrants, Fubini, the Steinhaus theorem on difference sets and the Lebesgue density theorem place a unit-area triangle with controlled angles at a density point, then the implicit function theorem (Lemma 3) perturbs it into a cyclic quadrilateral. For Theorem 2 it calls purely geometric observations sufficient: the argument concerns the region below the hyperbola 4xy = 1 and rests on Lemma 4. For Erdős's remark that the parallelogram version fails, the paper cites Section 6 of Kovač's earlier paper (arXiv:2309.09973); it proves no parallelogram result of its own. A progress-update section (Section 1.1, p. 2) reports that Koizumi, inspired by the proof of Theorem 1, then showed that every planar set of infinite measure contains the vertices of an isosceles triangle, a right triangle and an isosceles trapezoid of area 1, and says that this, with Theorems 1 and 2, resolves all five of Erdős's questions, which are jointly formulated as problem 353.

Source: https://arxiv.org/abs/2412.11725.

Bears on. #353: Theorem 1 answers yes the problem's cyclic-quadrilateral question and Theorem 2 answers no its question on convex polygons with congruent sides, for any number of sides. The paper proves nothing on the isosceles-trapezoid, isosceles-triangle and right-triangle questions, which it reports as settled by Koizumi.

Results. Labels and pages are those of arXiv:2412.11725v2.

• Theorem 1 (p. 1): every measurable planar set of infinite Lebesgue measure contains the four vertices of a non-degenerate cyclic quadrilateral of area 1; rescaling gives any prescribed area a > 0 (p. 1).
• Theorem 2 (p. 2), with Lemma 4 (p. 9): the set x > 1, y > 0, 4xy < 1 has infinite Lebesgue measure, and every convex polygon with congruent sides and all vertices in it has area strictly less than 1.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.