Andrzej P. Kisielewicz

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.

Rigid polyboxes and keller's conjecture

Abstract A cube tiling of ℝd is a family of axis-parallel pairwise disjoint cubes [0,1)d + T = {[0,1)d+t : t ∈ T} that cover ℝd. Two cubes [0,1)d + t, [0,1)d + s are called a twin pair if their closures have a complete facet in common. In 1930, Keller conjectured that in every cube tiling of ℝd there is a twin pair. Kellers conjecture is true for dimensions d ≤ 6 and false for all dimensions d ≥ 8. For d = 7 the conjecture is still open. Let x ∈ ℝd, i ∈ [d], and let L(T, x, i) be the set of all ith coordinates ti of vectors t ∈ T such that ([0,1)d+t) ∩ ([0,1]d+x) ≠ ø and ti ≤ xi. Let r−(T) = minx∈ℝd max1≤i≤d|L(T,x,i)| and r+(T) = maxx∈ℝd max1≤i≤d|L(T,x,i)|. It is known that Kellers conjecture is true in dimension seven for cube tilings [0,1)7 + T for which r−(T) ≤ 2. In the present paper we show that it is also true for d = 7 if r+(T) ≥ 6. Thus, if [0,1)d + T is a counterexample to Kellers conjecture in dimension seven, then r−(T), r+(T) ∈ {3, 4, 5}.

Continuous selection theorems for nonlower semicontinuous multifunctions

Abstract A continuous selection theorem for multifunctions, with nonempty Kuratowski–Painleve lower limit, whose values are C -convex and L -convex subsets of C(S, R n ) and L ∞ (T, R n ) is presented.

Partitions and balanced matchings of an n-dimensional cube.

Abstract A non-empty set A ⊆ X = X 1 × ⋯ × X n is a box if A = A 1 × ⋯ × A n and A i ⊆ X i for each i ∈ [ n ] . Two boxes A , B ⊂ X are dichotomous if A i = X i ∖ B i for some i ∈ [ n ] . Using a cube tiling code of R n designed by Lagarias and Shor, a certain class of partitions of an n -dimensional cube into 2 n pairwise dichotomous boxes is constructed. Additionally, for every prime number n ≥ 3 perfect matchings of the graph of the unit cube [ 0 , 1 ] n with faulty vertices ( 0 , … , 0 ) and ( 1 , … , 1 ) in which the number of edges in every direction is the same are presented.

On the number of minimal partitions of a box into boxes

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