Saturday 01 March 2025
For centuries, mathematicians have been fascinated by the properties of shapes and their relationships. One long-standing question has been whether every measurable planar set of infinite Lebesgue measure contains the vertices of an isosceles trapezoid of unit area. This problem was first posed by Paul Erdős in the 1980s, and since then, it has been a topic of intense research.
Recently, mathematician Junnosuke Koizumi has made significant progress on this problem. In his paper, he provides an affirmative answer to Erdős’ question, showing that every measurable planar set of infinite Lebesgue measure indeed contains the vertices of an isosceles trapezoid of unit area.
To understand how Koizumi achieves this result, let’s first consider a simpler problem: whether every measurable planar set of positive Lebesgue measure contains the vertices of an isosceles triangle of unit area. This problem was solved by Kovač and Predojević in the 1990s, but their method doesn’t directly apply to the more general case.
Koizumi’s approach begins with a clever trick: he replaces the original set S with its subset SR, which has a nice property that allows him to construct an isosceles trapezoid of unit area. By using this trick, he shows that there exists some point p in S such that f(p) is also contained in S.
The construction of f and its properties are the key to Koizumi’s proof. He defines a C∞-function ψ(r) that maps the real line to itself, and then uses this function to define a C∞-diomorphism f: R2 \ D → R2 \ D. This map has some remarkable properties: it preserves area, and its Jacobian determinant is equal to 1.
Using these properties, Koizumi shows that for any point p in S, the quadrilateral formed by p, f(p), R−1f(p), and R−1p is an isosceles trapezoid of unit area. By applying this construction repeatedly, he eventually finds a point q in SR such that f(q) is also contained in SR.
This result has far-reaching implications for our understanding of geometric shapes and their relationships. It also opens up new avenues for research into the properties of measurable sets and their connections to geometry.
Cite this article: “Solving Erdős Isosceles Trapezoid Problem”, The Science Archive, 2025.
Mathematics, Geometry, Shapes, Isosceles Trapezoid, Unit Area, Lebesgue Measure, Infinite Measure, Measurable Sets, C∞-Functions, Diomorphism
