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Rolf Schneider - Convex bodies in singular relative positions
ROLF SCHNEIDER, Albert-Ludwigs-Universität, Math. Institut, Eckerstr. 1, D-79104 Freiburg, Germany | |
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.
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