Abstract. We show that the Voronoi conjecture is true for parallelohedra with simply connected δ-surface. Namely, we show that if the boundary of parallelohedron P remains simply connected after removing closed non-primitive faces of codimension 2, then P is affinely equivalent to a Dirichlet-Voronoi domain of some lattice. Also we construct the π-surface associated with a parallelohedron and give another condition in terms of homology group of the constructed surface. Every parallelohedron with simply connected δ-surface also satisfies the condition on homology group of the π-surface.1. The Voronoi conjecturecan be tiled by non-overlapping translates of P.A tiling by parallelohedra is called face-to-face if the intersection of any two copies of P is a face of both, and is called a non face-to-face otherwise. B. Venkov [18] and later independently P. McMullen [13] proved that if there is a non face-to-face tiling by P , then there is also a face-to-face tiling. It is clear that the face-to-face tiling by P is unique up to translation. We will denote this tiling as T (P ) or just T when the generating polytope of the tiling is obvious.In 1897 H. Minkowski [15] proved that every parallelohedron P is centrally symmetric, and all facets of P are centrally symmetric. Later Venkov [18] added the third necessary condition, he proved that the projection of P along any face of codimension 2 is a twodimensional parallelohedron, i.e., a parallelogram or a centrally symmetric hexagon. Also Venkov proved that these three conditions are sufficient for a convex polytope to be a parallelohedron. In 1980 McMullen [13] gave an independent proof that these three conditions are necessary and sufficient, see also [14] for acknowledgment of priority.The centers of all tiles of T (P ) form a d-dimension lattice Λ(P ). If the fundamental domain of Λ(P ) has volume 1, then the homothetic polytope 2P is centrally symmetric with respect to the origin, has volume 2 d , and contains no lattice points in the interior. This, by definition, means that 2P is an extremal body. Additional information about extremal bodies can be found in [9, Ch.2 §12].On the other hand, given a d-dimensional lattice Λ, the Dirichlet-Voronoi polytope P (Λ) is a parallelohedron; the Dirichlet-Voronoi polytope is the set of points that are closer to a given lattice point O than to any other point of Λ.Conjecture 1 (G. Voronoi, [19]). Any d-dimensional parallelohedron P is affinely equivalent to a Dirichlet-Voronoi polytope P (Λ ′ ) for some d-dimensional lattice Λ ′ .Voronoi's conjecture has been proved for several families of parallelohedra with special local combinatorial properties.Denote the set of all k-faces of a tiling T by T k , and the set of all k-faces of a polytope P by P k .Date: December 23, 2014.

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We study and classify proper q-colourings of the ℤ d lattice, identifying three regimes where different combinatorial behaviour holds. (1) When $q\le d+1$ , there exist frozen colourings, that is, proper q-colourings of ℤ d which cannot be modified on any finite subset. (2) We prove a strong list-colouring property which implies that, when $q\ge d+2$ , any proper q-colouring of the boundary of a box of side length $n \ge d+2$ can be extended to a proper q-colouring of the entire box. (3) When $q\geq 2d+1$ , the latter holds for any $n \ge 1$ . Consequently, we classify the space of proper q-colourings of the ℤ d lattice by their mixing properties.

Let µ be a Borel probability measure in R d . For a k-flat α consider the value infµ(H), where H runs through all half-spaces containing α. This infimum is called the half-space depth of α.Bukh, Matoušek and Nivasch conjectured that for every µ and every 0 ≤ k < d there exists a k-flat with the depth at least k+1 k+d +1 . The Rado Centerpoint Theorem implies a lower bound of 1 d +1−k (the Rado bound), which is, in general, much weaker. Whenever the Rado bound coincides with the bound conjectured by Bukh, Matoušek and Nivasch, i.e., for k = 0 and k = d − 1, it is known to be optimal.In this paper we show that for all other pairs (d,k) one can improve on the Rado bound. If k = 1 and d ≥ 3 we show that there is a 1-dimensional line with the depth at least 1 d + 1 3d 3 . As a corollary, for all (d,k) satisfying 0 < k < d − 1 there exists a k-flat with depth at least 1 d +1−k + 1 3(d +1−k) 3 .