Distance completeness, maximal chords, and constant width in spheres (Q2771036)

For a 2-dimensional unit sphere \(S^2\), the author discusses counterparts of some properties of convex sets in \(E^2\). Here are some examples.NEWLINENEWLINENEWLINELet \(D\) be a compact nonempty set in \(S^2\) and \(\Delta= \Delta(D)\) be its diameter. Points \(p,q\in D\) are called a \(\mathbf{diameter}\) \(\mathbf{pair}\) if the distance \(d(p,q)= \Delta\).NEWLINENEWLINENEWLINEIf \(\Delta<\pi\) and for every \(p\in\partial D\) there exists \(q\in D\) such that the pair \(p\), \(q\) is diameter, then the set \(D\) is called CDP (complete with respect to diameter pairs). It is called CDS (complete with respect to diameter segments) if it is CDP and if for every diameter pair \(p,q\in\partial D\) the shortest geodesic segment \(\overline{pq}\subset D\). (Say, the set \(V\) of vertices of a tetrahedron inscribed into \(S^2\subset E^3\) is CDP but not CDS.)NEWLINENEWLINENEWLINEThe author adjusts a definition of definitely convex set in a Riemannian space to his setting as follows. A compact set \(D\) in \(S^2\) with a nonempty interior is called \(\mathbf{definitely}\) \(\mathbf{convex}\) if \(\Delta(D)< \pi\) and if, for any \(p,q\in D\), the segment \(\overline{pq}\subset D\) and \(\overline{pq}\cap\partial D\subset\{p, q\}\).NEWLINENEWLINENEWLINEIn these terms, the author proves the following:NEWLINENEWLINENEWLINEProposition 1. Any CDP-set \(D\) of diameter \(\Delta(D)< \pi/2\) has a definitely convex extension \(\widehat D\supset D\) which is a CDS-set of the same diameter. Moreover, \(\partial D\subset\partial\widehat D\).NEWLINENEWLINENEWLINEProposition 2. If \(D\) is a CDS-set of diameter \(\Delta(D)< \pi/2\), then \(D\) is definitely convex.NEWLINENEWLINENEWLINEThe author considers also complete sets in \(S^2\) which he calls SMD (saturated with respect to maximal distance). I.e. a set \(D\subset S^2\) is said to be SMD if addition of any point \(p\in S^2\setminus D\) will increase the diameter of \(D\). In contrast to the Euclidean case, completeness in \(S^2\) does not characterize bodies of constant width (naturally defined). Say, the set \(V\) above is SMD but not a body of constant width.NEWLINENEWLINENEWLINEProposition 3. Let \(D\subset S^2\) and \(\Delta(D)< \pi/2\). Then the following are equivalent:NEWLINENEWLINENEWLINEa) \(D\) is of constant width.NEWLINENEWLINENEWLINEb) \(D\) is an SMD-set.NEWLINENEWLINENEWLINEc) \(D\) is a CDS-set.