53 research outputs found
On Helly number for crystals and cut-and-project sets
We prove existence of finite Helly numbers for crystals and for
cut-and-project sets with convex windows; also we prove exact bound of
for the Helly number of a crystal consisting of copies of a single lattice.
We show that there are sets of finite local complexity that do not have finite
Helly numbers
On \pi-surfaces of four-dimensional parallelohedra
We show that every four-dimensional parallelohedron P satisfies a recently
found condition of Garber, Gavrilyuk & Magazinov sufficient for the Voronoi
conjecture being true for P. Namely we show that for every four-dimensional
parallelohedron P the group of rational first homologies of its \pi-surface is
generated by half-belt cycles.Comment: 16 pages, 7 figure
Belt distance between facets of space-filling zonotopes
For every d-dimensional polytope P with centrally symmetric facets we can
associate a "subway map" such that every line of this "subway" corresponds to
set of facets parallel to one of ridges P. The belt diameter of P is the
maximal number of line changes that you need to do in order to get from one
station to another.
In this paper we prove that belt diameter of d-dimensional space-filling
zonotope is not greater than . Moreover we show
that this bound can not be improved in dimensions d at most 6.Comment: 17 pages, 5 figure
Symmetries of Monocoronal Tilings
The vertex corona of a vertex of some tiling is the vertex together with the
adjacent tiles. A tiling where all vertex coronae are congruent is called
monocoronal. We provide a classification of monocoronal tilings in the
Euclidean plane and derive a list of all possible symmetry groups of
monocoronal tilings. In particular, any monocoronal tiling with respect to
direct congruence is crystallographic, whereas any monocoronal tiling with
respect to congruence (reflections allowed) is either crystallographic or it
has a one-dimensional translation group. Furthermore, bounds on the number of
the dimensions of the translation group of monocoronal tilings in higher
dimensional Euclidean space are obtained.Comment: 26 pages, 66 figure
Weighted cut-and-project sets in bounded distance to a lattice
Recent results of Grepstad and Lev are used to show that weighted
cut-and-project sets with one-dimensional physical space and one-dimensional
internal space are bounded distance equivalent to some lattice if the weight
function is continuous on the internal space, and if is either
piecewise linear, or twice differentiable with bounded curvature.Comment: 11 pages, 1 figur
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