Arrangements of homothets of a convex body

TL;DR

This paper provides upper bounds on the maximum number of pairwise intersecting homothets of a convex body in high dimensions, extending previous results to non-symmetric bodies and arbitrary interior points.

Contribution

It establishes new bounds for the number of intersecting homothets of convex bodies, including non-symmetric cases and arbitrary interior points, answering a question from 1994.

Findings

Maximum number of intersecting homothets is at most O(3^d d log d) for symmetric bodies.

Extended bounds to non-symmetric bodies with centroid as center.

Constructed families of at least Omega((2/√3)^d) translates with the property.

Abstract

Answering a question of F\"uredi and Loeb (1994), we show that the maximum number of pairwise intersecting homothets of a $d$-dimensional centrally symmetric convex body $K$, none of which contains the center of another in its interior, is at most $O(3^d d\log d)$. If $K$ is not necessarily centrally symmetric and the role of its center is played by its centroid, then the above bound can be replaced by $O(3^d\binom{2d}{d}d\log d)$. We establish analogous results for the case where the center is defined as an arbitrary point in the interior of $K$. We also show that in the latter case, one can always find families of at least $\Omega((2/\sqrt{3})^d)$ translates of $K$ with the above property.