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

✍ Scribed by V.C. Dumir; R.J. Hans-Gill; J.B. Wilker

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 965 KB
• Volume: 59
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1996.0102

✦ Synopsis

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 &(C, L) and for each dimension d, 1 d n&1, the relevant expansion factors are used to determine a supremum

Here a method for obtaining upper bounds on &(C, L) for ``rational'' subspaces L is given. This leads to many interesting results, e.g. it follows that the suprema &(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 &(B, n&2) and a Markoff type chain of related isolations for spheres B in R n with 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 &(C, L) and &(C, d ) are obtained.