Abstract
We show that for any concave polygon that has no parallel sides and for any k, there is a k-fold covering of some point set by the translates of this polygon that cannot be decomposed into two coverings. Moreover, we give a complete classification of open polygons with this property. We also construct for any polytope (having dimension at least three) and for any k, a k-fold covering of the space by its translates that cannot be decomposed into two coverings.
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Pálvölgyi, D. Indecomposable Coverings with Concave Polygons. Discrete Comput Geom 44, 577–588 (2010). https://doi.org/10.1007/s00454-009-9194-y
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DOI: https://doi.org/10.1007/s00454-009-9194-y