A problem of blocking light rays
Let \(K\) be an \(n\)-dimensional convex body with interior \(\text{int} (K)\). Let \(g(K)\) (resp. \(h(K))\) be the smallest number of non-overlapping translates of \(K\) (resp. \(\text{int} (K))\) which block all the light rays starting from \(K\). Denote by \(g^*(K)\) and \(h^*(K)\) similar numbers when the translates are taken from lattice packings of \(K\). The author proves the following results: Theorem 1: For every \(n\)-dimensional convex body \(K\) we have \(g(K)\geq 2n\), where the equality holds if and only if \(K\) is a parallelepiped. The result for \(n=2\) had been obtained by Grünbaum in 1961. Theorem 2: For every \(n\)-dimensional convex body \(K\) we have \(g(K)\leq (8e)^n (n+1)^{n-1} n^{3(n^2-1)/2}\). For every \(n\)-dimensional centrally symmetric convex body \(C\) we have \(g(C)\leq (8e)^n (n+1)^{n-1} n^{n^2-1}\). Theorem 3: When \(n\geq 3\), there is an \(n\)-dimensional convex body \(K\) such that \(g^*(K) <\infty\) but \(h^*(K) =\infty\). Theorem 4: For every two-dimensional convex domain \(D\), \(h^*(D)=12\).
scientific article; zbMATH DE number 1089231