On convex and starshaped hulls (Q2718045)

In the first 2 paragraphs, the authors prove some known or easy to see basic properties of convex sets. An example isNEWLINENEWLINENEWLINELemma 1.1. Let \(S\) be a compact subset of \(E^n\), \(n\geq 2\). Then the convex hull of \(S\) is the intersection of all closed half-spaces supporting \(S\) (i.e., containing \(S\) and bounded by supporting hyperplanes).NEWLINENEWLINENEWLINEAnother example isNEWLINENEWLINENEWLINETheorem 2.2. Let \(B\subset E^n\) be a compact connected set with a non-empty interior. If \(x\in \text{conv} B\), then each hyperplane through \(x\) has a non-empty intersection with \(B\).NEWLINENEWLINENEWLINEThe authors follow the proof of Theorem 2.2 by a remark that ``if \(B\) has no interior points, then the above theorem is no longer true.'' This remark is wrong: Theorem 2.2 obviously holds even if \(\text{int} B\) is empty. NEWLINENEWLINENEWLINEIn \S 3, for \(A\subset E^n\), \(p\in A\), the authors introduce the starshaped hull \(SH_p(A)\). In essence, this is the union of all segments from \(p\) to other points of \(A\). Their basic observation here is that if \(SH_p(A)\) is convex, then \(SH_p(A)= \text{conv} A\).