The view-obstruction problem for 3-dimensional spheres

Abstract

LetC be ann-dimensional sphere with diameter 1 and center at the origin inE ⁿ. The view-obstruction problem forn-dimensional spheres is to determine a constant ν(n) to be the lower bound of those α for which any half-lineL, given byx _i =a _i t (i=1,2,...,n) where parametert≥0 anda _i (i=1,2,...,n) are positive real numbers, intersects

$$\Delta (C,\alpha ) = \left\{ {\alpha C + \left( {m_1 + \frac{1}{2},m_2 + \frac{1}{2},...,m_n + \frac{1}{2}} \right):m_1 ,m_2 ,...,m_n nonnegative integers} \right\}$$

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In this paper, forn=3, the following result is proved. For α >$1/\sqrt 2$ we have that any half-lineL, given byx _i =a _i t(i=1,2,3), intersects Δ(C,α), where parametert≥0 anda _i (i=1,2,3) are positive real numbers such that |a|+|b|+|c|≠3 wheneveraa ₁+ba ₂+ca ₃=0 for three integersa,b,c.