On projective hedgehogs (envelopes parametrized by their Gauss map)
As a generalization of the fact that a closed strictly convex hypersurface \(M^n\) in Euclidean space \(\mathbb{R}^{n+ 1}\) may be considered as envelope of its support hyperplanes \(\langle x,p\rangle= h(p)\) \((p\in S^n)\), \textit{R. Langevin}, \textit{G. Levitt} and \textit{H. Rosenberg} [Banach Cent. Publ. 20, 245-253 (1988; Zbl 0658.53004)] denoted such an envelope \({\mathcal H}_h\) by ``hedgehog'' (hérisson) if its defining function \(h: S^n\to\mathbb{R}^n\) is any function of class \({\mathcal C}^\infty\). \({\mathcal H}_h\) may be parametrized by \(S^n\) and has the representation \(x_h(p)= h(p)p+ (\text{grad }h)(p)\). In general, there exist singularities for \({\mathcal H}_h\); and in the case of \(h(-p)= -h(p)\), the parametrization may be reduced to the projective space \(P^n\) (projective hedgehog). The author investigates projective hegehogs as central asymmetries of convex hedgehogs, in particular of convex hedgehogs of constant width. A typical result of him is the fact that every analytical projective hedgehog is situated in the convex hull of its singularities. Finally, he presents a hedgehog version of Steiner's Roman surface in \(\mathbb{R}^3\) admitting an axial symmetry like Boy's surface.
scientific article; zbMATH DE number 1121327