View-obstruction problems. II

Abstract

Let S n {S^n} denote the region 0 > x i > ∞ ( i = 1 , 2 , … , n ) 0 > {x_i} > \infty (i = 1,2, \ldots ,n) of n n -dimensional Euclidean space E n {E^n} . Suppose C C is a closed convex body in E n {E^n} which contains the origin as an interior point. Define α C \alpha C for each real number α ⩾ 0 \alpha \geqslant 0 to be the magnification of C C by the factor α \alpha and define C + ( m 1 , … , m n ) C + ({m_1}, \ldots ,{m_n}) for each point ( m 1 , … , m n ) ({m_1}, \ldots ,{m_n}) in E n {E^n} to be the translation of C C by the vector ( m 1 , … , m n ) ({m_1}, \ldots ,{m_n}) . Define the point set Δ ( C , α ) \Delta (C,\alpha ) by Δ ( C , α ) = { α C + ( m 1 + 1 2 , … , m n + 1 2 ) : m 1 , … , m n \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 C is the problem of finding the constant K ( C ) K(C) defined to be the lower bound of those α \alpha such that any half-line L L given by x i = a i t ( i = 1 , 2 , … , n ) {x_i} = {a_i}t(i = 1,2, \ldots ,n) , where the a i ( 1 ⩽ i ⩽ n ) {a_i}(1 \leqslant i \leqslant n) are positive real numbers, and the parameter t t runs through [ 0 , ∞ ) [0,\infty ) , intersects Δ ( C , α ) \Delta (C,\alpha ) . The paper considers the case where C C is the n n -dimensional cube with side 1, and in this case the constant K ( C ) K(C) is known for n ⩽ 3 n \leqslant 3 . The paper gives a new proof for the case n = 3 n = 3 . Unlike earlier proofs, this one could be extended to study the cases with n ⩾ 4 n \geqslant 4 .