A projective characterisation of ellipsoids by isotropy (Q2474154)

A set \(M\) of the real projective \(3\)-space \(P^3\) is called convex, if \(M\) is a convex subset of a three-dimensional affine subspace \(A^3\subset P^3\). A convex set \(M\subset P^3\) is said to be isotropic with respect to the point \(Q\in M\), if for each pair \(A, B\) of the boundary \(\partial M\) of \(M\) there exists a (projective) collineation of \(M\) onto itself such that \(Q\mapsto Q\) and \(A\mapsto B\). The author proves the following ``Projective isotropy theorem'': If \(M\) is a convex open subset of \(P^3\), which is isotropic with respect to some point \(Q\in M\), then either \(M=A^3\) or \(M\) is an ellipsoid. An analogous theorem is valid for the real projective plane \(P^2\).