Strictly Convex Simplexwise Linear Embeddings of a 2-Disk

Let$K \subset R^2$be a finitely triangulated 2-disk; a map f: K → R2is called simplexwise linear (SL) if f ∣ σ is affine linear for each (closed) 2-simplex σ of K. Let$E(K) = \{\text{orientation preserving} \mathrm{SL} \text{embeddings} K \rightarrow R^2\}, E_{\mathrm{sc}}(K) = \{f \in E(K) \mid f(K...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 1985-01, Vol.288 (2), p.723-737
Main Author: Bloch, Ethan D.
Format: Article
Language:English
Subjects:
Circles
Convexity
Embeddings
Line segments
Mathematical theorems
Moving images
Topological spaces
Vertices
Online Access:Get full text
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Summary:Let$K \subset R^2$be a finitely triangulated 2-disk; a map f: K → R2is called simplexwise linear (SL) if f ∣ σ is affine linear for each (closed) 2-simplex σ of K. Let$E(K) = \{\text{orientation preserving} \mathrm{SL} \text{embeddings} K \rightarrow R^2\}, E_{\mathrm{sc}}(K) = \{f \in E(K) \mid f(K) \text{is strictly convex}\}$, and let$\overline{E(K)}$and$\overline{E_{\mathrm{sc}}(K)}$denote their closures in the space of all$\mathrm{SL} \text{maps} K \rightarrow R^2$. A characterization of certain elements of$\overline{E(K)}$is used to prove that Esc(K) has the homotopy type of S1and to characterize those elements of$\overline{E(K)}$which are in$\overline{E_{\mathrm{sc}}(K)}$, as well as to relate such maps to SL embeddings into the nonstandard plane.
ISSN:0002-9947
DOI:10.1090/S0002-9947-1985-0776400-3