Fractional illumination and the optimal exponential rate in Hadwiger's covering conjecture

We show that the fractional illumination number of every convex body in ${\mathbb{R}}^d$ is at most $2^d$, with equality exactly for parallelotopes. We also prove that every such body can be covered by at most $2^d(d\log d+d\log \log d+O(d))$ smaller positive homothetic copies as $d\to \mathrm{\infty }$, establishing the optimal exponential rate in Hadwiger’s covering conjecture. The proofs use a covering measure obtained by minimizing an overlap energy and a greedy covering argument on a finite net.