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

TL;DR

This paper investigates polygons of unit area with vertices in sets of infinite planar measure, providing answers to questions about cyclic quadrilaterals and convex polygons with congruent sides, advancing understanding in geometric measure theory.

Contribution

It offers the first positive and negative answers to longstanding questions about the existence of certain polygons within infinite measure sets.

Findings

Cyclic quadrilaterals of area 1 can be found in every infinite measure set.

Convex polygons with congruent sides of area 1 cannot always be found in such sets.

Abstract

Paul Erd\H{o}s 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.