Convex and discrete geometry Convex geometric analysis and geometric inequalities Problem

Illumination problem

Builds on Convex body Boundary Interior Hadwiger covering conjecture

Problem

Let $n \ge 1$ , and let $K \subseteq \R^n$ be a convex body. A nonzero vector $u$ illuminates a boundary point $x$ of $K$ if the ray $\set{x + t u : t > 0}$ meets the interior of $K$ . The illumination number $I(K)$ is the least number of vectors such that every boundary point of $K$ is illuminated by one of them. Show that $I(K) \le 2^n$ , with equality only when $K$ is a parallelotope.

Remarks

Boltyanski proved that $I(K)$ equals the least number of smaller homothetic copies $x_i + \lambda K$ , with $0 < \lambda < 1$ , that cover $K$ . So the problem is equivalent to the Hadwiger covering conjecture, and it is open for $n \ge 3$ . Example: for a disc, a vector $u$ illuminates the boundary point with outer normal $\nu$ exactly when $u \cdot \nu < 0$ . Three vectors at angles $2\pi/3$ illuminate every boundary point, while two vectors never suffice, so $I = 3$ . For the square, each corner needs its own direction, so $I = 4 = 2^2$ .

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Boundary concept
Convex body concept
Homothety concept
Interior concept
Hadwiger covering conjecture conjecture
Illumination problem problem

Tag	`J1E6BEFAPZ`
URI	`m2://J1E6BEFAPZ`
Kind	problem
Writing	W0 stub
MSC	52A20, 52C17
Status	This copy of the site has no build data. Run `m2 status J1E6BEFAPZ`.
About	Convex body
Uses	Convex body, Boundary, Interior, Hadwiger covering conjecture

Illumination problem

Problem

Remarks

Reading path

Further reading

What this page needs for W1

What links here