Polygons of unit area with vertices in sets of infinite planar measure

TL;DR

The paper addresses unit-area finite configurations within planar sets of infinite measure. It proves that every infinite-measure set contains four concyclic points forming a cyclic quadrilateral of area $1$, using a measure-theoretic construction and a local implicit-function argument. It also exhibits an explicit infinite-measure set for which no convex polygon with congruent sides can attain area $1$, establishing a negative result for that variant. Collectively, these results resolve several Erdős–Mauldin questions about unit-area polygons in dense planar sets and illuminate the qualitative geometric structure available in infinite-measure contexts.

Abstract

Paul Erdős and R. Daniel Mauldin asked a series of questions on certain types of polygons of area $1$, the vertices of which can be found in every planar set of infinite Lebesgue measure. We address two of these questions, one on cyclic quadrilaterals and the other on convex polygons with congruent sides, with respectively positive and negative answers.

Figures (9)

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Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Lemma 3
• proof : Proof of Lemma \ref{['lm:thetriangle']}
• Lemma 4
• proof : Proof of Lemma \ref{['lm:hyparea']}