Shadow Boundaries of Convex Bodies

If C is a convex body in R^n and X is a k-dimensional linear subspace of R^n, we denote by S(C,X) the shadow boundary of C over X which is defined as the collection of all points which belong to C and to one of its tangent (n-k)-flats orthogonal to X. For almost all directions in R^3, the shadow boundary is a curve encompassing the body C. It has been established long ago by G. Ewald, D.G. Larman and C.A. Rogers [11] that, for every given C, S(C,X) is almost always a topological (k-1)-sphere. As a follow on from this result, in 1974 Peter McMullen asked whether most of these shadow boundaries would have finite “length” [15]. This is already shown to be true for polytopes and also true for general convex bodies when the dimension of the subspace X is 1 or n-1. Here we show that almost all shadow boundaries have finite “length” whatever the dimension k, 0< k< n, of the subspace X. The set of shadow boundaries of infinite “length” has also been considered in the context of Baire category. In 1989, P. Gruber and H. Sorger proved that, in the Baire category sense, most pairs (C,X), where C is a convex body in R^n and X an (n-1)-dimensional subspace of R^n, produce shadow boundaries S(C,X) of infinite length. Here we show that this result also holds for pairs (C,X) where X is a k-dimensional subspace, 0< k< n. We also consider the length of increasing paths in the 1-skeleton of a convex body. We conclude with observations and open questions arising from the work on shadow boundaries of the first two chapters