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arXiv · 1608.04639

Arrangements of homothets of a convex body

Márton Naszódi·János Pach·Konrad Swanepoel
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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.

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BibTeXRIS

Márton Naszódi, János Pach, Konrad Swanepoel. 2016-08-16. Arrangements of homothets of a convex body. https://doi.org/10.1112/s0025579317000122

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