Hedgehogs and area of order 2
We define an algebraic area of order 2 for hedgehogs (envelopes parametrized by their Gauss map) in \(E^{n+1}\). On the vector space of hedgehogs defined up to a translation, this yields a nondegenerate quadratic form that allows us to define a scalar product on some subspaces, such as the subspace of projective hedgehogs. As an application, we give some characterizations of spheres among all convex hedgehogs of constant width. Finally, we prove that for minimal hedgehogs of finite area in \(E^3\) the square root of the area is a convex function of the support function, and we extend this result to asymptotic areas of embedded ends.
envelopes parametrized by their Gauss map
scientific article; zbMATH DE number 938588