The complex Illumination problem

We formulate a complex analog of the celebrated Levi-Hadwiger-Boltyanski illumination (or covering) conjecture for complex convex bodies in Cn, as well as its (non-comparable) fractional version. A key element in posing these problems is computing the classical and fractional illumination numbers of the complex analog of the hypercube, i.e., the polydisc. We prove that the illumination number of the polydisc in Cn is equal to 2n+1-1 and that the fractional illumination number of the polydisc in Cn is equal to 2n. In addition, we verify both conjectures for the classes of complex zonotopes and zonoids.