Hadwiger’s covering (illumination) conjecture

Determine whether the Hadwiger covering numbers H_n and H_s equal 2^n for every integer n ≥ 3. Concretely, establish that for every n-dimensional convex body K, the minimal number H_n of translates of int(K) needed to cover K is 2^n, and that the corresponding minimal number H_s for centrally symmetric convex bodies is also 2^n.

Background

Hadwiger’s covering conjecture (also called the illumination conjecture) asserts a sharp universal bound of 2^n for covering any n-dimensional convex body by translates of its interior, with analogous expectation for centrally symmetric bodies. The paper defines the covering numbers H_n (general convex bodies) and H_s (centrally symmetric convex bodies) and focuses on improving upper bounds in small dimensions, while noting that the conjecture itself remains unresolved.

The authors provide new explicit bounds for H_n and H_s in low dimensions and discuss known asymptotic bounds, but the global equality H_n = H_s = 2^n is still not established.