Two convex bodies K and K′ in Euclidean space [ ]n can be said to be in exceptional relative position if they have a common boundary point at which the linear hulls of their normal cones have a non-trivial intersection. It is proved that the set of rigid motions g for which K and gK′ are in exceptional relative position is of Haar measure zero. A similar result holds true if ‘exceptional relative position’ is defined via common supporting hyperplanes. Both results were conjectured by S. Glasauer; they have applications in integral geometry.