Intersecting ellipses induced by a max-sum matching

TL;DR

This paper proves that for a set of points in the plane, the convex sets bounded by ellipses with foci at max-sum matching edges always intersect, answering a longstanding geometric question.

Contribution

It introduces a novel geometric proof that convex sets from max-sum matchings intersect, solving a Tverberg-type problem posed in 1995.

Findings

Convex sets bounded by ellipses with specific foci intersect

Max-sum matchings induce intersecting geometric structures

Answers a 1995 open question in discrete geometry

Abstract

For an even set of points in the plane, choose a max-sum matching, that is, a perfect matching maximizing the sum of Euclidean distances of its edges. For each edge of the max-sum matching, consider the ellipse with foci at the edge's endpoints and eccentricity $\sqrt 3 / 2$. Using an optimization approach, we prove that the convex sets bounded by these ellipses intersect, answering a Tverberg-type question of Andy Fingerhut from 1995.