Simple polygonal regions, diagonals, and triangulations

A simple polygonal region is a compact connected set $P\subseteq {\mathbb{R}}^2$ such that $\mathrm{int}P$ is nonempty and connected, $P=\overline{\mathrm{int}P}$, and $\mathrm{\partial }P$ is the union of the edges of an irredundant simple closed finite polygonal chain.

Explicitly, the boundary chain has distinct cyclic vertices $v_0,\dots ,v_{n-1}$ with $n\ge 3$. With indices read modulo $n$, its closed edges are $[v_i,v_{i+1}]$. Nonconsecutive edges are disjoint, consecutive edges meet only at their common endpoint, and no three consecutive vertices are collinear. Compactness, interior, closure, boundary, and connectedness are taken in the Euclidean metric $d_2(x,y)=\parallel x-y{\parallel }_2$ of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and ${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it, with the notions of Open cover, subcover, compact metric space, and compact subset of a metric space, Interior, closure, boundary, limit point, isolated point and dense subset of a metric space, and Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets; finiteness is that of The cardinality $\mid A\mid$ of a finite set.

A diagonal $[v_i,v_j]$ joins two nonadjacent boundary vertices and its open segment $\{(1-t)v_i+t{v}_j:0<t<1\}$ lies in $\mathrm{int}P$.

A triangulation is a finite family of nondegenerate closed triangles with union $P$ such that the intersection of any two distinct triangles is empty, a common vertex, or a full common edge. It is frugal when the set of all triangle vertices is exactly the boundary-vertex set of $P$; a general triangulation may also use finitely many subdivision vertices on boundary edges or in the interior.

This definition begins with an already given filled set $P$ and imposes conditions on it. It does not assert that every simple closed polygonal chain determines such a set. The boundary-chain convention is called a polygon in Polygonal arcs and polygons as non-self-intersecting finite unions of line segments in ${\mathbb{R}}^2$ ↗, and Polygonal Jordan curve theorem: a polygon has exactly two complementary regions and is the frontier of each ↗ supplies the corresponding complementary-region theorem.