Convex hulls of generalized moment curves
In this nice paper, the author determines the facial structure of the convex hull \(C_{pq},\{p,q\}\subset {\mathbb{Z}}^+\), of the curves \(M_{pq}=\{z(t)=z_{pq}(t)=(\cos 2\pi pt,\sin 2\pi pt,\cos 2\pi qt,\sin 2\pi qt)| \in {\mathbb{R}}\}.\) The proper faces of \(C_{pq}\) include a) the 0-faces: \(\{\) z(t)\(\}\), \(0\leq t<1\), b) the 1-faces: \(\{[z(s),z(t)]| 0\leq s<t<1,t-s\in (\frac{k}{p},\frac{\ell}{q})\cup (1-\frac{\ell}{q},1-\frac{k}{p})\}\) where \(1\leq p<q\), \(0\leq k<p\), \(1\leq \ell <q\) and \(\ell p-kq=1,\) but c) there are no 3-faces. In addition, conv \(\{\) z(t\(+\frac{i}{p})| 0\leq i<p-1\), \(0\leq t<\frac{1}{p}\}\) and \(conv \{z(t+\frac{i}{q})| 0\leq i\leq q-1,0\leq t<\frac{1}{q}\}\) are proper faces of \(C_{pq}\) which are 2-faces respectively when \(p\geq 3\) and \(q\geq 3\), and 1-faces respectively when \(p=2\) and \(q=2\).
facial structure of the convex hull of trigonometric curves
scientific article; zbMATH DE number 3935874