Illumination of direct sums of two convex figures
Let a subset \(M\) of \(\mathbb R^n\) be a compact convex body. A boundary point \(x\) is illuminated by a vector \(a\neq 0\) if, for any sufficiently small \(\lambda>0\), the point \(x+\lambda a\) lies in int\(M\). Denote by \(c(M)\) the least integer \(c\) for which there exist \(c\) nonzero vectors \(a_1, a_2,\dots, a_c\) which illuminate the whole boundary of \(M\). The Hadwiger conjecture is that \(c(M)\leq 2n\). This is trivial for \(n=1\) and has been proved for \(n=2\). For \(n\geq 3\), only partial results are known. The authors prove for \(n=4\) and \(M\) being a direct sum of two compact convex 2-dimensional bodies that \(c(M)\) can be only 7, 8, 9, 12, and 16.
scientific article; zbMATH DE number 5117807