On affinely embeddable sets in the projective plane (Q2276761)

A cone \(C\subset {\mathbb{R}}^ 3\) is called pointed if it contains no line. Theorem: Assume \(n\geq 3\) and \(C_ 1,...,C_ n\subset {\mathbb{R}}^ 3\) are closed pointed, convex cones with common apex the origin 0. Assume that for \(i\neq j\) \((i,j=1,2,...,n)\) there is an \(e(i,j)\in \{-1,+1\}\) such that for all \(k=1,...,n\), \(k\neq i,j\) and for both \(e=1\) and \(e=-1\) we have \[ (eC_ k)\cap (C_ 1+e(i,j)C_ j)=\{0\}. \] Then there is a plane P through 0 such that for all \(i=1,...,n\), \(P\cap C_ j=\{0\}\). The theorem is translated to the projective plane \(P^ 2\) proving a conjecture of \textit{T. Bisztriczky} and \textit{J. Schaer} [Acta Math. Hung. 49, 353-363 (1987; Zbl 0625.52002)]. According to the author the extension to higher dimensional spaces is possible but the conditions in the theorem become rather unintelligible.