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View-obstruction problems. II. (English) Zbl 0474.10023

Summary: [Part I, cf. Aequat. Math. 9, 165–170 (1973; Zbl 0265.52003).]
Let \(S^n\) denote the region \(0< x_i < \infty\) \((i = 1, 2, \ldots, n)\) of \(n\)-dimensional Euclidean space \(E^n\). Suppose \(C\) is a closed convex body in \(E^n\) which contains the origin as an interior point. Define \(\alpha C\) for each real number \(\alpha\ge 0\) to be the magnification of \(C\) by the factor \(\alpha\) and define \(D+ (m_1, \ldots, m_n)\) for each point \((m_1, \ldots, m_n)\) in \(E^n\) to be the translation of \(C\) by the vector \((m_1, \ldots, m_n)\). Define the point set \(\Delta(C,\alpha)\) by \(\Delta(C,\alpha) = \{\alpha C + (m_1 + \tfrac12, \ldots, m_n + \tfrac12): m_1, \ldots, m_n\) nonnegative integers \(\}\).
The view-obstruction problem for \(C\) is the problem of finding the constant \(K(C)\) defined to be the lower bound of those \(\alpha\) such that any half-line \(L\) given by \(x_i = a_it\) \((i = 1,2, \ldots, n)\), where the \(a_i\) \((1\le i\le n)\) are positive real numbers, and the parameter \(t\) runs through \([0,\infty)\), intersects \(\Delta(C,\alpha)\). The paper considers the case where \(C\) is the \(n\)-dimensional cube with side \(1\), and in this case the constant \(K(C)\) is known for \(n\le 3\).
The paper gives a new proof for the case \(n=3\). Unlike earlier proofs, this one could be extended to study the cases with \(n\ge 4\).

MSC:

11J99 Diophantine approximation, transcendental number theory
11H06 Lattices and convex bodies (number-theoretic aspects)
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A30 Variants of convex sets (star-shaped, (\(m, n\))-convex, etc.)

Citations:

Zbl 0265.52003
Full Text: DOI

References:

[1] U. Betke and J. M. Wills, Untere Schranken für zwei diophantische Approximations-Funktionen, Monatsh. Math. 76 (1972), 214 – 217 (German). · Zbl 0239.10016 · doi:10.1007/BF01322924
[2] T. W. Cusick, View-obstruction problems, Aequationes Math. 9 (1973), 165 – 170. · Zbl 0265.52003 · doi:10.1007/BF01832623
[3] T. W. Cusick, View-obstruction problems in \?-dimensional geometry, J. Combinatorial Theory Ser. A 16 (1974), 1 – 11. · Zbl 0273.10025
[4] I. J. Schoenberg, Extremum problems for the motions of a billiard ball. II. The \?_{\infty } norm, Nederl. Akad. Wetensch. Proc. Ser. A 79=Indag. Math. 38 (1976), no. 3, 263 – 279. · Zbl 0357.90077
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