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

Abstract

The $$\mu $$ μ -kernel of an o-symmetric convex body is obtained by shrinking the body about its center by a factor of $$\mu $$ μ . As a generalization of pairwise intersecting Minkowski arrangement of o-symmetric convex bodies, we can define the pairwise intersecting Minkowski arrangement of order $$\mu $$ μ . Here, the homothetic copies of an o-symmetric convex body are so that none of their interiors intersect the $$\mu $$ μ -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 $$3^d$$ 3 d . The case $$\mu =1$$ μ = 1 is the Bezdek–Pach Conjecture, which asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in $$\mathbb R^d$$ R d is $$2^d$$ 2 d . We verify the conjecture on the plane, that is, when $$d=2$$ d = 2 . Indeed, we show that the number in question is four for any planar convex body.

On generalized Minkowski arrangements

“…Minkowski arrangements consisting of congruent convex bodies were considered in [4]. Estimates for the maximum cardinality of mutually intersecting members in a (generalized) Minkowski arrangement can be found in [11,15,16,18]. The problem investigated in this paper is similar in nature to those dealing with the volume of the convex hull of a family of convex bodies, which has a rich literature.…”

On generalized Minkowski arrangements

“…Minkowski arrangements consisting of congruent convex bodies were considered in [4]. Estimates for the maximum cardinality of mutually intersecting members in a (generalized) Minkowski arrangement can be found in [11,14,15,17]. The problem investigated in this paper is similar in nature to those dealing with the volume of the convex hull of a family of convex bodies, which has a rich literature.…”