Rolf Schneider - Convex bodies in singular relative positions

A common boundary point x of the convex bodies K, K' in Euclidean n-space is exceptional if the linear hulls of the normal cones of K and K' at x have a non-zero intersection. A common support hyperplane H of K and K' is exceptional if the affine hulls of the support sets and have a nonempty intersection or contain parallel lines.

Theorem 1. The set of all rigid motions g for which K and gK' have some exceptional common boundary point is of Haar measure zero.

Theorem 2. The set of all rigid motions g for which K and gK' have some exceptional common support hyperplane is of Haar measure zero.

These results continue a line of research on the boundary structure of convex bodies that started with a question of Victor Klee. The proof applies the `cap covering method' due to Ewald, Larman and Rogers. Both theorems were conjectured by Stefan Glasauer; they are needed in integral-geometric investigations on generalized curvature measures of convex bodies.