# Difference between revisions of "Illumination problem"

The problem of determining the minimum number of directions of pencils of parallel rays, or number of sources, illuminating the whole boundary of a convex body. Let $K$ be a convex body in an $n$- dimensional linear space $\mathbf R ^ {n}$, let $\mathop{\rm bd} K$ and $\mathop{\rm int} K$ be respectively its boundary and its interior, and assume that $\mathop{\rm bd} K \neq \emptyset$. The best known illumination problems are the following.

1) Let $l$ be a certain direction in $\mathbf R ^ {n}$. A point $x \in \mathop{\rm bd} K$ is called illuminated from the outside by the direction $l$ if the straight line passing through $x$ parallel to $l$ passes through a certain point $y \in \mathop{\rm int} K$ and if the direction of the vector $\vec{xy}$ coincides with $l$. The minimum number $c ( K)$ of directions in the space $\mathbf R ^ {n}$ is sought that is sufficient to illuminate the whole set $\mathop{\rm bd} K$.

2) Let $z$ be a point of $\mathbf R ^ {n} \setminus K$. A point $x \in \mathop{\rm bd} K$ is called illuminated from the outside by the point $z$ if the straight line defined by the points $z$ and $x$ passes through a point $y \in \mathop{\rm int} K$ and if the vectors $\vec{xy}$ and $\vec{zy}$ have the same direction. The minimum number $c ^ \prime ( K)$ of points from $\mathbf R ^ {n} \setminus K$ is sought that is sufficient to illuminate the whole set $\mathop{\rm bd} K$.

3) Let $z$ be a point of $\mathop{\rm bd} K$. A point $x \in \mathop{\rm bd} K$ is illuminated from within by the point $z \neq x$ if the straight line defined by the points $z$ and $x$ passes through a point $y \in \mathop{\rm int} K$ and if the vectors $\vec{xy}$ and $\vec{zy}$ have opposite directions. The minimum number $p( K)$ of points from $\mathop{\rm bd} K$ is sought that is sufficient to illuminate the whole set $\mathop{\rm bd} K$ from within.

4) A system of points $Z = \{ {z } : {z \in \mathop{\rm bd} K } \}$ is said to be fixing for $K$ if it possesses the properties: a) $Z$ is sufficient to illuminate the whole set $\mathop{\rm bd} K$ from within; and b) $Z$ does not have any proper subset sufficient to illuminate the set $\mathop{\rm bd} K$ from within. The maximum number $p ^ \prime ( K)$ of points of a fixing system is sought for the body $K \subset \mathbf R ^ {n}$.

Problem 1) was proposed in connection with the Hadwiger hypothesis (see [1]): The minimum number of bodies $b( K)$ homothetic to a bounded $K$ with homothety coefficient $k$, $0< k< 1$, sufficient for covering $K$, satisfies the inequality $n+ 1 < b( K) \leq 2 ^ {n}$, whereby the value $b( K) = 2 ^ {n}$ characterizes a parallelepiped. For $K \subset \mathbf R ^ {n}$ bounded, $c( K) = b( K)$. If $K$ is unbounded, then $c( K) \leq b( K)$, and there exist bodies such that $c( K) < b( K)$ or $c( K) = b( K) = \infty$( see [1]).

Problem 2) was proposed in connection with problem 1). For $K \subset \mathbf R ^ {n}$ bounded, the equality $c( K) = c ^ \prime ( K)$ holds. If $K$ is not bounded, then $c ^ \prime ( K) \leq b( K)$ and $c( K) \leq c ^ \prime ( K)$. The number $c ^ \prime ( K)$ for any unbounded $K \subset \mathbf R ^ {3}$ takes one of the values 1, 2, 3, 4, $\infty$( see [1]).

The solution of problem 3) takes the form: The number $p( K)$ is defined if and only if $K$ is not a cone. In this case,

$$2 \leq p( K) \leq n+ 1 ,$$

whereby $p( K) = n+ 1$ characterizes an $n$- dimensional simplex of the space $\mathbf R ^ {n}$( see [1]).

For problem 4) (see [2]), it has been conjectured that if $K \subset \mathbf R ^ {n}$ is bounded, the inequality

$$p ^ \prime ( K) \leq 2 ^ {n}$$

holds.

Every illumination problem is closely linked to a special covering of the body $K$( cf. Covering (of a set)) (see [1]).

#### References

 [1] V.G. Boltyanskii, P.S. Soltan, "The combinatorial geometry of various classes of convex sets" , Kishinev (1978) (In Russian) [2] B. Grünbaum, "Fixing systems and inner illumination" Acta Math. Acad. Sci. Hung. , 15 (1964) pp. 161–163