Hadwiger conjecture (combinatorial geometry)

Unsolved problem in mathematics:

Can every ${\displaystyle n}$-dimensional convex body be covered by ${\displaystyle 2^{n}}$ smaller copies of itself?

(more unsolved problems in mathematics)

In combinatorial geometry, the Hadwiger conjecture states that any convex body in n-dimensional Euclidean space can be covered by 2ⁿ or fewer smaller bodies homothetic with the original body, and that furthermore, the upper bound of 2ⁿ is necessary if and only if the body is a parallelepiped. There also exists an equivalent formulation in terms of the number of floodlights needed to illuminate the body.

The Hadwiger conjecture is named after Hugo Hadwiger, who included it on a list of unsolved problems in 1957; it was, however, previously studied by Levi (1955) and independently, Gohberg & Markus (1960). Additionally, there is a different Hadwiger conjecture concerning graph coloring—and in some sources the geometric Hadwiger conjecture is also called the Levi–Hadwiger conjecture or the Hadwiger–Levi covering problem.

The conjecture remains unsolved even in three dimensions, though the two dimensional case was resolved by Levi (1955).