An illumination problem for zonoids (Q1261891)

For a convex body \(K\) in Euclidean space \(E^ d\) let \(I_ 1(K)\) be the smallest number \(m\) so that \(K\) can be illuminated by \(m\) lines in \(E^ d\backslash K\). The authors conjecture that \[ I_ 1(K)\leq\left\lceil{2^ d\over d+1}\right\rceil, \] and they prove this for the special case where \(K\) is a zonoid and \(d+1=2^ p\), \(p\geq 2\) an integer.