An algebraic contribution to the conic theorem of G. J. Rieger. (Q1182570)

This text reviews both the paper by \textit{G. J. Rieger} [ibid. 35, No. 2, 263-270 (1988) with a complement: ibid. 36, No. 2, 244-246 (1989); see the papers below] and the underlying paper by G. Pickert. For any function of \([-a,a]\) into \(\mathbb{R}\) with suitable regularity condition and \(F(0)=1\) a cone is defined as a set \(K_ F=\{(zt,zF(t),z); | t| \leq a\}\). G. J. Rieger considers functions \(F\) such that \(K_ F\cap E(k)\) and \(K_ F\wedge E(-k)\) are congruent, where \(E(k)\) is the plane \(\{(x,y,z)\mid y=1+kz\}\); more explicitly, the intersections are translates along the \(z\)-axis. The argument proceeds with power series and results in the function \(F\) being of the form \(\sqrt{1+ct^ 2}\). The postscript gives a simplification of the argument eliminating the parameters earlier. The author takes up the same question in a homogenized fashion considering \(K=\{(zx_ 0,zy_ 0,z)\mid z\geq 0, (x_ 0,y_ 0)\in C\}\) with \((0,1)\in C\), \((0,1,1)\in K\), with a translation condition and an initial condition. He defines a group of projective mappings and shows that \(C\) is a union of orbits under this group.