Bounds on convex bodies in pairwise intersecting Minkowski arrangement of order mu

Földvári, Viktória (2020) Bounds on convex bodies in pairwise intersecting Minkowski arrangement of order mu. Journal of Geometry, 111 (2). ISSN 0047-2468, ESSN: 1420-8997

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Abstract

The mu-kernel of an o-symmetric convex body is obtained by shrinking the body about its center by a factor of mu. As a generalization of pairwise intersecting Minkowski arrangement of o-symmetric convex bodies, we can define the pairwise intersecting Minkowski arrangement of order mu. Here, the homothetic copies of an o-symmetric convex body are so that none of their interiors intersect the mu-kernel of any other. We give general upper and lower bounds on the cardinality of such arrangements, and study two special cases: For d-dimensional translates in classical pairwise intersecting Minkowski arrangement we prove that the sharp upper bound is 3d. The case mu = 1 is the Bezdek-Pach Conjecture, which asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in Rd is 2d. We verify the conjecture on the plane, that is, when d = 2. Indeed, we show that the number in question is four for any planar convex body.

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