Contributions to a general theory of view-obstruction problems, II

Abstract

In view-obstruction problems, congruent copies of a closed, centrally symmetric, convex body C, centred at the points of the shifted lattice ( ½, ½, ...,½ )+ Zⁿin Rⁿ, are expanded uniformly. The expansion factor required to touch a given subspace L is denoted by v(C,L) and for each dimensiond, 1≤d≤n-1, the relevant expansion factors are used to determine a supremum v(C,d)=sup {v(C,L): dimL=d,L not contained in a coordinate hyperplane}. Here a method for obtaining upper bounds on v(C,L) for "rational" subspaces L is given. This leads to many interesting results, e.g. it follows that the suprema v(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 obtain v(B,n-2) and a Markoff type chain of related isolations for spheres Bin Rⁿwith n≥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 for v(C,L) and v(C,d) are obtained.