View-Obstruction Problems. II

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 $n \geqslant 4$.