Illumination conjecture for general convex bodies

Prove the illumination conjecture that every compact convex body in \(\mathbb{R}^n\) can be illuminated by a set of at most \(2^n\) points.

Background

The paper identifies the classical illumination conjecture as an unresolved problem in convex and discrete geometry. The conjecture concerns arbitrary compact convex bodies in $\mathbb{R}^n$, in contrast to the smooth-boundary setting studied in the paper, for which the illumination number is known to be $n+1$. The paper addresses an effective lattice-restricted variant for smooth convex bodies rather than proving the general conjecture.