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

2024/12/16 by Kovač, Vjekoslav, Predojević, Bruno
#28A75 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2412.11725

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.

Related

• Isosceles trapezoids of unit area with vertices in sets of infinite planar measure
• The area of cyclic polygons: Recent progress on Robbins' Conjectures
• A Set of Questions in Combinatorial and Metric Geometry
• An algorithm to find maximum area polygons circumscribed about a convex\n polygon
• Polygons in hyperbolic geometry 2: Maximality of area