The Complex Illumination Problem

TL;DR

This paper extends the classical illumination conjecture to complex convex bodies, computes illumination numbers for complex polydiscs, and verifies the conjectures for complex zonotopes and zonoids.

Contribution

It formulates a complex analog of the illumination conjecture, computes exact illumination numbers for complex polydiscs, and verifies the conjectures for specific complex convex bodies.

Findings

Illumination number of complex polydisc in C^n is 2^(n+1)-1

Fractional illumination number of complex polydisc in C^n is 2^n

Conjectures verified for complex zonotopes and zonoids

Abstract

We formulate a complex analog of the celebrated Levi-Hadwiger-Boltyanski illumination (or covering) conjecture for complex convex bodies in C^n, 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 C^n is equal to 2^(n+1)-1 and that the fractional illumination number of the polydisc in C^n is 2^n. In addition, we verify both conjectures for the classes of complex zonotopes and zonoids.