Arrangements of homothets of a convex body
M\'arton Nasz\'odi, J\'anos Pach, Konrad Swanepoel

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 -dimensional centrally symmetric convex body , none of which contains the center of another in its interior, is at most . If is not necessarily centrally symmetric and the role of its center is played by its centroid, then the above bound can be replaced by . We establish analogous results for the case where the center is defined as an arbitrary point in the interior of . We also show that in the latter case, one can always find families of at least translates of with the above property.
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