A note on Kuperberg's quadrilateral conjecture
Abstract
Kuperberg conjectured in 1983 that every convex body K in the plane is contained in a quadrilateral of area at most 3/√5 |K|, the extremal bodies being the affine-regular pentagons. On the basis of numerical experiments we propose a stronger conjecture of a purely polygonal nature: every convex polygon P is contained in a quadrilateral of area at most 3/√5 |P| whose sides are parallel to sides or diagonals of P. This \emph{chord conjecture} implies the inequality conjectured by Kuperberg. For pentagons it is a theorem of Hong, Ismailescu, Kwak and Park, of which we give a short proof by area identities. We also show that the classical bound √2 remains valid for quadrilaterals with sides parallel to chords; the proof is similar to Ismailescu's proof of this bound, but it starts from an inscribed quadrilateral of maximal area instead of a circumscribed quadrilateral of minimal area. We close with a discussion of the difficulties in reaching the constant 3/√5.
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BibTeX
@misc{vigh2026note,
title = {{A note on Kuperberg's quadrilateral conjecture}},
author = {Viktor Vígh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08168},
eprint = {2610.08168},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08168},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A note on Kuperberg's quadrilateral conjecture
AU - Viktor Vígh
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08168
UR - https://arxiv.org/abs/2610.08168
AB - Kuperberg conjectured in 1983 that every convex body K in the plane is contained in a quadrilateral of area at most 3/√5 |K|, the extremal bodies being the affine-regular pentagons. On the basis of numerical experiments we propose a stronger conjecture of a purely polygonal nature: every convex polygon P is contained in a quadrilateral of area at most 3/√5 |P| whose sides are parallel to sides or diagonals of P. This \emph{chord conjecture} implies the inequality conjectured by Kuperberg. For pentagons it is a theorem of Hong, Ismailescu, Kwak and Park, of which we give a short proof by area identities. We also show that the classical bound √2 remains valid for quadrilaterals with sides parallel to chords; the proof is similar to Ismailescu's proof of this bound, but it starts from an inscribed quadrilateral of maximal area instead of a circumscribed quadrilateral of minimal area. We close with a discussion of the difficulties in reaching the constant 3/√5.
ER -