Strictly convex simplexwise linear embeddings of a $2$-disk

Abstract:

Let $K \subset {{\mathbf {R}}^2}$ be a finitely triangulated $2$-disk; a map $f:K \to {{\mathbf {R}}^2}$ is called simplexwise linear $(SL)$ if $f|\sigma$ is affine linear for each (closed) $2$-simplex $\sigma$ of $K$. Let $E(K) = \{ {\text {orientation preserving SL embeddings}}\;K \to {{\mathbf {R}}^2}\}$, ${E_{{\text {SC}}}}(K) = \{ f \in E(K)|f(K)\;{\text {is strictly convex}}\}$, and let $\overline {E(K)}$ and $\overline {{E_{{\text {SC}}}}(K)}$ denote their closures in the space of all ${\text {SL}}$ maps $K \to {{\mathbf {R}}^2}$. A characterization of certain elements of $\overline {E(K)}$ is used to prove that ${E_{{\text {SC}}}}(K)$ has the homotopy type of ${S^1}$ and to characterize those elements of $\overline {E(K)}$ which are in $\overline {{E_{{\text {SC}}}}(K)}$, as well as to relate such maps to ${\text {SL}}$ embeddings into the nonstandard plane.