Illumination Problem
Ask how many directions are needed to illuminate every boundary point of a convex body.
Hadwiger's illumination conjecture asks for the minimum number of external directions (or, equivalently, smaller homothetic copies) needed to cover every boundary point of a convex body in n-dimensional space. The conjecture states that this number is at most 2^n, with equality if and only if the body is a parallelotope. Levi resolved the two-dimensional case in 1955, showing that four directions always suffice and only parallelograms require all four. In three dimensions, Prymak proved in 2023 that 14 translates suffice, still short of the conjectured 8; the general conjecture remains open for all n >= 3.
Formula
The fewest directions of light needed so that every boundary point of a convex body K receives light.
The conjecture bounds the illumination number by 2ⁿ, with equality for parallelepipeds (cubes, boxes) only.
The illumination number equals the minimum number of smaller homothetic copies of K needed to cover K.
Summary
Hadwiger's illumination conjecture asks for the minimum number of external directions (or, equivalently, smaller homothetic copies) needed to cover every boundary point of a convex body in n-dimensional space. The conjecture states that this number is at most 2^n, with equality if and only if the body is a parallelotope. Levi resolved the two-dimensional case in 1955, showing that four directions always suffice and only parallelograms require all four. In three dimensions, Prymak proved in 2023 that 14 translates suffice, still short of the conjectured 8; the general conjecture remains open for all n >= 3.
