On the existence of topological ovals in flat projective planes
Any closed oval (= smooth convex set) in a compact, connected projective plane is a topological oval; it is homeomorphic to a sphere. Such ovals exist only in \(2\)- and \(4\)-dimensional planes [\textit{H. Salzmann, D. Betten, Th. Grundhöfer, H. Hähl, R. Löwen} and \textit{M. Stroppel}, `Compact projective planes', Berlin : de Gruyter (1995; Zbl 0851.51003), Theorem 55.14]. The tangents of a closed oval form an oval in the dual plane. All known examples in \(2\)-dimensional (= flat) non-Desarguesian planes consist of the absolute points of a polarity (and hence are analogous to conics). By a clever construction, the authors show that in any flat plane \(\mathcal P\) each triangle is contained in a topological oval. A set \(\mathcal Q\) of flags in \(\mathcal P\) is called a quasi-oval, if it is contained in a suitable affine subplane of \(\mathcal P\) such that for each flag \((p,L)\) in \(\mathcal Q\) the other points of \(\mathcal Q\) are contained in one half-plane with respect to \(L\). Any finite quasi-oval can be extended successively in such a way that the closure of the point set is a full oval.
scientific article; zbMATH DE number 1028213