The View-Obstruction Problem for 3-Dimensional Spheres

Let C be an n-dimensional sphere with diameter 1 and center at the origin in En. The view-obstruction problem for n-dimensional spheres is to determine a constant v（n） to be the lower bound of those α for which any half-line L, given by xi=ait（i=1, 2,...,n） where parameter t≥0 and ai（i=1, 2,...,n） are positive real numbers, intersects Δ（C, α）={αC+（m1+（1/2）, m2+（1/2）,…,mn+（1/2））:m1, m2,…mn nonnegative integers}. In this paper, for n=3, the following result is proved. For α】1/51/2 we have that any half-line L, given by xi=ait（i=1,2,3）, intersects Δ（C, α）, where parameter t≥0 and ai（i=1,2,3） are positive real numbers such that |a|+|b|+|c|≠3 whenever aa1+ba2+ca3=0 for three integers a, b, c.