View-obstruction problem for 3-dimensional spheres
The view-obstruction problem for an n-dimensional closed convex body C (containing the origin in its interior) has been formulated by \textit{T. W. Cusick} [Aequationes Math. 9, 165-170 (1973; Zbl 0265.52003)]. The problem is to determine the constant K(C) defined to be the infimum of those \(\alpha >0\), such that for any ray L starting from the origin and lying in the positive quadrant, there are non-negative integers \(m_ 1,...,m_ n\) for which L intersects \(\alpha C+(m_ 1+,...,m_ n+)\). K(C) is denoted by \(\nu\) (n) when C is the n-dimensional sphere with centre 0 and diameter 1. Here it is shown that \(\nu (3)=\sqrt{3/7}\) thereby proving a conjecture of Cusick (loc. cit.). It is further shown that the result is isolated and a complete chain of isolated minima converging to 1/\(\sqrt{3}\) is obtained. The constant 1/\(\sqrt{3}\) is needed for uncountably many rays. In the proof, the authors use isolation results for the view-obstruction problem for circles and a variant (see lemmas 1 and 2). The authors expect that analogous isolation results hold for the cube with centre 0 in \({\mathbb{R}}^ 3\).
scientific article; zbMATH DE number 3943930
scientific article