Hedgehogs and zonoids (Q5937629)

For a real function \(f\) of class \({\mathcal C}^2\) on the unit sphere \(\mathbb{S}^n\) of Euclidean space \((\mathbb{R}^{n+1}, \langle\cdot, \cdot\rangle)\), the hedgehog with support function \(f\) is the parametrized surface (in general with singularities) \(x_f:\mathbb{S}^n \to\mathbb{R}^{n+1}\) with \(x_f(u)= f(u)u+ (\text{grad} f)(u)\); thus it represents the envelope, denoted by \(H_f\), of the family of hyperplanes given by \(\langle x,u\rangle= f(u)\). Let \(R_f\) be the product of the principal radii of curvature of \(x_f\) and define NEWLINE\[NEWLINEh(u): ={1\over 2} \int_{ \mathbb{S}^n} \bigl|\langle u,v\rangle \bigr|R_f(v)d \sigma(v), \quad u\in \mathbb{R}^nNEWLINE\]NEWLINE \((\sigma=\) spherical Lebesgue measure). The author shows that \(h\) is of class \({\mathcal C}^2\), and hence \(h|\mathbb{S}^n\) defines a hedgehog \(H_h=: \Pi_f \). In the case where \(f\) is the restriction to \(\mathbb{S}^n\) of a sublinear function on \(\mathbb{R}^n\), \(H_f\) is the boundary of a convex body, and \(\Pi_f\) is the projection body of this body. The author extends some results from the theory of convex projection bodies to these new projection hedgehogs \(\Pi_f\). He also obtains some results on the classical projection bodies (zonoids). Example: Let \(K\) be a zonoid whose generating measure has a continuous density with respect to spherical Lebesgue measure. If one of the principal radii of curvature is zero at \(p\), then all its principal radii of curvature are zero at \(p\).