Krzysztof Przesławski

Polyboxes, Cube Tilings and Rigidity

Abstract A non-empty subset A of X=X1×⋅⋅⋅×Xd is a (proper) box if A=A1×⋅⋅⋅×Ad and Ai⊂Xi for each i. Suppose that for each pair of boxes A, B and each i, one can only know which of the three states takes place: Ai=Bi, Ai=Xi∖Bi, Ai∉{Bi,Xi∖Bi}. Let F and G be two systems of disjoint boxes. Can one decide whether ∪F=∪G? In general, the answer is ‘no’, but as is shown in the paper, it is ‘yes’ if both systems consist of pairwise dichotomous boxes. (Boxes A, B are dichotomous if there is i such that Ai=Xi∖Bi.) Several criteria that enable to compare such systems are collected. The paper includes also rigidity results, which say what assumptions have to be imposed on F to ensure that ∪F=∪G implies F=G. As an application, the rigidity conjecture for 2-extremal cube tilings of Lagarias and Shor is verified.

On the structure of cube tilings and unextendible systems of cubes in low dimensions

We give a structural description of cube tilings and unextendible cube packings of R 3 . We also prove that up to dimension 4 each cylinder of a cube tiling contains a column and demonstrate by an example that the latter result does not hold in dimension 5.

Decomposability of Polytopes

A known characterization of the decomposability of polytopes is reformulated in a way which may be more computationally convenient, and a more transparent proof is given. New sufficient conditions for indecomposability are then deduced, and illustrated with some examples.

On the number of minimal partitions of a box into boxes

The number of minimal partitions of a box into proper boxes is examined.