View-obstruction problems in n-dimensional geometry

Abstract

Let S n denote the region 0 < x i < ∞ ( i = 1,2,…, 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 αC for each real number α ⩾ 0 to be the set of all ( αx 1,…, αx n ), where ( x 1,…, x n ) is a point in C. Define C + ( m 1,…, m n ) for each point ( m 1,…, m n ) of E n to be the set of all ( x 1 + m 1,…, x n + m n ), where ( x 1,…, x n ) is a point in C. Define the point set Δ( C, α) by Δ(C, α) = {αC + (m 1 + 1 2 ,…,m n + 1 2 ): m 1,…,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 i t ( 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.