Abstract
LetK n ⊂ ℝn be a triangulatedn-ball. Examples are given to show that unlike in the two-dimensional case, the following hold for alln≥3: (1) there are nonconvexK n with no convex simplexwise linear embeddingsK n → ℝn, even though there are strictly convex simplexwise linear embeddings ∂K n → ℝn; (2) there are convexK n, with no spanning simplices, such that not every simplexwise linear embeddingf: ∂K n → ℝn with convex image can be extended to a simplexwise linear embedding ofK n; (3) there are convexK n such that the space of simplexwise linear homeomorphisms ofK n, fixed on ∂K n, is not path connected.
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Partially supported by NSF Contract DMS-8503388.
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Bloch, E.D. Complexes whose boundaries cannot be pushed around. Discrete Comput Geom 4, 365–374 (1989). https://doi.org/10.1007/BF02187737
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DOI: https://doi.org/10.1007/BF02187737