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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



View-obstruction problems. II

Author: T. W. Cusick
Journal: Proc. Amer. Math. Soc. 84 (1982), 25-28
MSC: Primary 10F10; Secondary 52A43, 52A45
MathSciNet review: 633270
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Abstract: 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 \geqslant 0$ to be the magnification of $ C$ by the factor $ \alpha $ and define $ C + ({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} + \frac{1} {2}, \ldots ,{m_n} + \frac{1} {2}):{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_i}t(i = 1,2, \ldots ,n)$, where the $ {a_i}(1 \leqslant i \leqslant 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 \leqslant 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 \geqslant 4$.

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Article copyright: © Copyright 1982 American Mathematical Society