Mauldin 2013 some problems ideas erdos analysis geometry

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section_5: Mauldin's Section 5: the triangle-of-area-one question as Problems 5.1 and 5.2, the stated equivalence with finite unions of convex interiors, and the argument that the conjectured constant 4π/(3√3) is best possible for unions of at most three such interiors.

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R. Daniel Mauldin, Some problems and ideas of Erdős in analysis and geometry, in Erdős Centennial, Bolyai Soc. Math. Stud. 25 (2013), 365--376; DOI 10.1007/978-3-642-39286-3_13.

The copy read for this card is the author's TeX preprint of the chapter, headed "Department of Mathematics, University of North Texas, Denton" and dated January 28, 2013, eleven pages numbered 1--11 with a text layer (253,662 bytes). The printed chapter (pp. 365--376) was not compared, so the page numbers below are the preprint's. Provenance: downloaded in September 2026; the download URL was not recorded. The preprint prints no copyright or license line (pp. 1-2 and 10-11 read); its download URL was not recorded, so no hosting page was consulted, and the publisher's chapter page describes the printed chapter, not the preprint; the term is unstated.

Read status. Claims checked: Section 5 (preprint pp. 6--8) was read in full on the page images and its statements were compared clause by clause with the question on the problem page; its arguments were read but not verified. The rest of the preprint was read on the page images for its statements only.

Contents

The items below are the parts of Section 5, "Sets containing the vertices of a triangle of area 1", that concern the problem; the Section 5 result page states them.

• Opening remark (p. 6): Erdős long ago noted that if $E\subseteq\mathbb R^2$ is Lebesgue measurable with infinite measure, then for every $c>0$ the set $E$ contains the vertices of a triangle of area $c$; the paper adds that several people have noted that this remains true if $E$ has positive measure and is unbounded.
• Problem 5.1 (p. 7), attributed on p. 6 to Erdős's [17] (Real Anal. Exchange 1978/79), [4] (his Scottish Book problems, 1981) and [14] (Oberwolfach 1983): is there a finite constant $C$ such that every Lebesgue measurable set $E$ of measure greater than $C$ contains the vertices of a triangle of area $1$? Moreover, is the best constant $c_0=4\pi/(3\sqrt3)$, the area of the disk whose inscribed equilateral triangle has area $1$? Mauldin adds that, as far as Mauldin knows, Erdős never offered money for it. The problem page writes "measure $\ge c$"; the difference does not affect the existence of a finite constant.
• Problem 5.2 (p. 7): the equivalent form, obtained "using some standard approximations in measure theory" (not spelled out), in which $E$ is the union of the interiors of at most $n$ compact convex sets and one finite constant $c$ must serve every $n\in\mathbb N$; it again asks whether $c_0$ is the best possible constant.
• Partial results (pp. 7--8): $c_0$ is the best possible constant when $n=1$ (Steiner symmetrizations of a convex body containing no triangle of area greater than $1$ converge to a disk of the same area; from the author's [29], Mauldin 2002, and repeated here), when $n=2$ (if the union of two compact convex bodies contains the vertices of no triangle of area greater than $1$, neither does their convex hull; cited to [29]), and when $n=3$ (a "redistribution of mass" argument written out on pp. 7--8: a "small" triple reduces to $n=1$ through its convex hull, and for a "large" triple the region swept out between the two larger bodies has area at least that of the smallest, so the three bodies can be replaced by one). The general case is left open.

Compiled scope

Section 5 was read in full. Sections 2--4 and 6--7 (similar copies of sequences, additive number theory and effective dimension, dimension of subgroups and rings, partitions of lines and planes, exact dimension of continued fractions using only the primes) were read for their statements only; they restate results of other papers, among them Lorentz's theorem, the Erdős--Kunen--Mauldin theorem, the Erdős--Volkmann theorem and the Erdős--Jackson--Mauldin partition theorem, and open problems, and record no result of this paper, so they have no result page here. No statement here is independently reviewed, and the published chapter was not compared with the preprint.

Bears on. #352: Section 5 records the problem with Erdős's conjectured constant $c_0$ and states its equivalence with the finite-union-of-convex-interiors form; its argument for unions of the interiors of at most three compact convex sets answers the question for those sets, with any constant greater than $c_0$, and shows that no smaller constant serves there. It leaves the general question open.

Result. Section 5 (pp. 6--8).

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