Contributions to a General Theory of View-Obstruction Problems, II

Abstract

In view-obstruction problems, congruent copies of a closed, centrally symmetric, convex bodyC, centred at the points of the shifted lattice (12, 12, ..., 12)+Znin Rn, are expanded uniformly. The expansion factor required to touch a given subspaceLis denoted byν(C,L) and for each dimensiond, 1≤d≤n−1, the relevant expansion factors are used to determine a supremumν(C,d)=sup {ν(C,L): dimL=d,Lnot contained in a coordinate hyperplane}.Here a method for obtaining upper bounds onν(C,L) for “rational” subspacesLis given. This leads to many interesting results, e.g. it follows that the supremaν(C,d) are always attained and a general isolation result always holds. The method also applies to give simple proofs of known results for three dimensional spheres. These proofs are generalized to obtainν(B,n−2) and a Markoff type chain of related isolations for spheresBin Rnwithn≥4. In another part of the paper, the subspaces occurring in view-obstruction problems are generalized to arbitrary flats. This generalization is related to Schoenberg's problem of billiard ball motion. Several results analogous to those forν(C,L) andν(C,d) are obtained.