Dumir, V. C. ; Hans-Gill, R. J. ; Wilker, J. B.
(1996)
*Contributions to a general theory of view-obstruction problems, II*
Journal of Number Theory, 59
(2).
pp. 352-373.
ISSN 0022-314X

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/S00223...

Related URL: http://dx.doi.org/10.1006/jnth.1996.0102

## 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 )+ 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 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 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 v(C,L) and v(C,d) are obtained.

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Deposited On: | 10 Nov 2010 05:42 |

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