Rotors in triangles and tethrahedra

Abstract

A polytope $P$ is circumscribed about a convex body $\Phi \subset {\mathbb{R}}^n$ if $\Phi \subset P$ and each facet of $P$ is contained in a support hyperplane of $\Phi$. We say that a convex body $\Phi \subset {\mathbb{R}}^n$ is a rotor of a polytope $P$ if for each rotation $\rho$ of ${\mathbb{R}}^n$ there exist a translation $\tau$ so that $P$ is circumscribed about $\tau \rho \Phi$. In this paper we shall prove that if $P$ is a triangle, then there is a baricentric formula that describes the curvature of bd$\Phi$ at the contact points, $\{A_1,A_2,A_3\}$. We prove also that if $\Phi \subset {\mathbb{R}}^3$ is a convex body which is a rotor in a tetrahedron $T$ and if $\Phi$ intersects the faces of $T$ at the points $\{x_1,\dots ,x_4\}$, then the normal lines of $\Phi$ at the contact points with $T$, $\{x_1,\dots ,x_4\}$ generically belong to one ruling of a quadric surface.