Let \(N\geq0\) and let \(v_0,\ldots,v_k\in\RR^N\) be affinely independent, meaning that \(v_1-v_0,\ldots,v_k-v_0\) are linearly independent. The \(k\)-simplex \([v_0,\ldots,v_k]\subseteq\RR^N\) is the convex hull of \(v_0,\ldots,v_k\), and these points are its vertices. A face of \([v_0,\ldots,v_k]\) is a simplex spanned by a nonempty subset of \(\ts{v_0,\ldots,v_k}\); for \(0\leq i\leq k\) and \(k\geq1\), the \(i\)-th facet is \[ \del_i[v_0,\ldots,v_k]\coloneqq[v_0,\ldots,\hat v_i,\ldots,v_k]. \] A simplicial complex in \(\RR^N\) is a set \(K\) of simplices in \(\RR^N\) such that:
every face of a simplex in \(K\) belongs to \(K\);
for \(\sigma,\tau\in K\), the intersection \(\sigma\cap\tau\) is either empty or a face of both \(\sigma\) and \(\tau\);
every point of \(\RR^N\) has a neighborhood that meets only finitely many simplices of \(K\).