View-obstruction problems in n-dimensional geometry

Let Sⁿ denote the region 0 < x_i < ∞ (i = 1,2,...,n) of n-dimensional Euclidean space Eⁿ. Suppose C is a closed convex body in Eⁿ which contains the origin as an interior point. Define αC for each real number α ≥ 0 to be the set of all (αx₁,...,αx_n), where (x₁,...,x_n) is a point in C. Define C + (m₁,...,m_n) for each point (m₁,...,m_n) of Eⁿ to be the set of all (x₁ + m₁,...,x_n + m_n), where (x₁,...,x_n) is a point in C. Define the point set Δ(C, α) by Δ(C, α) = {αC + (m₁ + 1 2,...,m_n + 1 2): m₁,...,m_n non-negative integers}. The view-obstruction problem for C is the problem of finding the constant K(C) defined to be the lower bound of those α such that any half-line L given by x_i = a_it (i = 1,2,...,n), where the a_i(1 ≤ i ≤ n) are positive real numbers and the parameter t runs through [0, ∞], intersects Δ(C, α). The simplest choices for C are the n-dimensional cube with side 1 and the n-dimensional sphere with diameter 1: let λ(n) and ν(n), respectively, denote the constant K(C) in these cases. Elementary geometry is enough to prove that λ(2) = 1 3 and ν(2) = 1 √5. The paper uses a method partly analytic and partly combinatorial to prove that λ(3) = 1 2. The conjecture λ(n) = (n -1) (n + 1) for each n ≥ 2 is stated, and a connection with a certain Diophantine approximation problem is shown.