Bounds on Convex Bodies in Pairwise Intersecting Minkowski Arrangement of Order $\mu$

TL;DR

This paper establishes bounds on the size of pairwise intersecting Minkowski arrangements of convex bodies, verifies the Bezdek-Pach Conjecture in the plane, and explores special cases with sharp bounds.

Contribution

It provides new upper and lower bounds for Minkowski arrangements and confirms the Bezdek-Pach Conjecture in two dimensions.

Findings

Upper bound of 3^d for d-dimensional translates in Minkowski arrangements.

Verification of the Bezdek-Pach Conjecture in the plane, showing the maximum is four.

Derived bounds and properties for arrangements involving the μ-kernel of convex bodies.

Abstract

A generalization of pairwise intersecting Minkowski arrangement of centrally symmetric convex bodies is the pairwise intersecting Minkowski arrangement of order $\mu$. Here, the homothetic copies of a centrally 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 $3^d$. For $\mu=1$ the general version yields to another known problem: The Bezdek-Pach Conjecture asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in $\mathbb{R}^d$ is $2^d$. 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.