The Complex Illumination Problem

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.

Figures (4)

• Figure 1: A visualization, for $n=3$, of the fact that if $x_3\neq -x_4$ then any $v\in V_1$ which illuminates $x_1a_1+x_2a_2\in K_1$ also illuminates $x=x_1a_1+x_2a_2+x_3a_3+x_4a_4\in K$. Here $Da_j=K(a_j)$ is the disc induced by $a_j$ and $z$ is a direction illuminating both $x_3\in Da_3$ and $x_4\in Da_4$. Using a small refinement by $\delta z$ and \ref{['eq:comp_zon_dep']} we write $x+tv=x+tv+\delta z(\lambda_1a_1+\dots+\lambda_{4}a_{4})$ in the form of \ref{['eq:comp_zon_int']}.
• Figure 2: A visualization, for $n=3$, of the fact that if $x_3=-x_4$ and $v\in V_2$ illuminates $x_1a_1+x_3a_3\in K_2$, then $v'\in V_3$ illuminates $x=x_1a_1+x_2a_2+x_3a_3+x_4a_4\in K$. Here $Da_j=K(a_j)$ is the disc induced by $a_j$ and $z$ is a direction which illuminates $x_3\in Da_3$. Using a small refinement by $\delta z$ and \ref{['eq:comp_zon_dep']} we can write $x+tv'=x+tv'+\delta z (\lambda_1a_1+\dots+\lambda_4 a_4)$ in the form of \ref{['eq:comp_zon_int']}.
• Figure 3: A representation of a complex zonotope $K=D_1+D_2+D_3\subseteq\mathbb{C}^2$ by three circles. Given a point $x=x_1a_1+x_2a_2+x_3a_3\in K_0$ and a direction $e^{\theta i}$ which is not orthogonal to any vector in $\{x_1,x_2,x_3\}$, the resulting segments $A_i=\ell^\theta_i\cap D_i$ are proper. Also notice that their sum $A$ can be translated to the centrally-symmetric real zonotope $A'$ (the sum of the segments $A'_1,A'_2,A'_3$ in grey), both of which are contained in $K$.
• Figure 4: $y_i$ will illuminate $x_i$ if $x_i$ is in the arc $ab$, with $a,b$ being the points of tangency corresponding to $y_i$. We have $|\arg y_i-\arg x_i|=\theta_x<\theta$,which can be calculated using $|oy_i|=r$.

Theorems & Definitions (53)

• Conjecture
• Conjecture 1.1
• Conjecture 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Definition 1.7
• Theorem 1.8
• Theorem 1.9