Brick Partition Problems In Three Dimensions

A d-dimensional brick is a set I-1 x ... x I-d where each I-i is an interval. Given a brick B, a brick partition of B is a partition of B into bricks. A brick partition P-d of a d-dimensional brick is k-piercing if every axis-parallel line intersects at least k bricks in Pd. Bucic et al. (2019) explicitly asked the minimum size p(d, k) of a k-piercing brick partition of a d-dimensional brick. The answer is known to be 4(k - 1) when d = 2. Our first result almost determines p(3, k). Namely, we construct a k-piercing brick partition of a 3-dimensional brick with 12k - 15 parts, which is off by only 1 from the known lower bound. As a generalization of the above question, we also seek the minimum size s(d, k) of a brick partition P-d of a d-dimensional brick where each axis-parallel plane intersects at least k bricks in P-d. We resolve the question in the 3-dimensional case by determining s(3, k) for all k. (C) 2021 Elsevier B.V. All rights reserved.