Abstract
Two dependent examples are presented: 1. Two convex polytopes in ℝ3 such that for each pair of their parallel facets, one of the facets fits strictly into the other. (The example gives a refinement of A. D. Alexandrov’s uniqueness theorem for convex polytopes.) 2. A pointed tiling of the two-sphere S2 generated by a Laman-plus-one graph which can be regularly triangulated without adding extra vertices.
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