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

References

[a1]	R. Schneider, "Boundary structure and curvature of convex bodies" J. Tölke (ed.) J.M. Wills (ed.) , Contributions to geometry , Birkhäuser (1979) pp. 13–59
[a2]	V. [V.G. Boltyanskii] Boltjansky, I. [I. Gokhberg] Gohberg, "Results and problems in combinatorial geometry" , Cambridge Univ. Press (1985) (Translated from Russian)