Contributions to a general theory of view-obstruction problems .2.

In view-obstruction problems, congruent copies of a closed, centrally symmetric, convex body C, centred at the points of the shifted lattice (1/2,1/2,...,1/2)+Z(n) in R(n), are expanded uniformly. The expansion factor required to touch a given subspace L is denoted by v(C, L) and for each dimension d, 1 less than or equal to d less than or equal to n-1, the relevant expansion factors are used to determine a supremum v(C, D)=sup{v(C, L): dim L=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 B in R(n) with n greater than or equal to 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. (C) 1996 Academic Press, Inc.