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 be a convex body in an -dimensional linear space , let and be respectively its boundary and its interior, and assume that . The best known illumination problems are the following.

1) Let be a certain direction in . A point is called illuminated from the outside by the direction if the straight line passing through parallel to passes through a certain point and if the direction of the vector coincides with . The minimum number of directions in the space is sought that is sufficient to illuminate the whole set .

2) Let be a point of . A point is called illuminated from the outside by the point if the straight line defined by the points and passes through a point and if the vectors and have the same direction. The minimum number of points from is sought that is sufficient to illuminate the whole set .

3) Let be a point of . A point is illuminated from within by the point if the straight line defined by the points and passes through a point and if the vectors and have opposite directions. The minimum number of points from is sought that is sufficient to illuminate the whole set from within.

4) A system of points is said to be fixing for if it possesses the properties: a) is sufficient to illuminate the whole set from within; and b) does not have any proper subset sufficient to illuminate the set from within. The maximum number of points of a fixing system is sought for the body .

Problem 1) was proposed in connection with the Hadwiger hypothesis (see [1]): The minimum number of bodies homothetic to a bounded with homothety coefficient , , sufficient for covering , satisfies the inequality , whereby the value characterizes a parallelepiped. For bounded, . If is unbounded, then , and there exist bodies such that or (see [1]).

Problem 2) was proposed in connection with problem 1). For bounded, the equality holds. If is not bounded, then and . The number for any unbounded takes one of the values 1, 2, 3, 4, (see [1]).

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

whereby characterizes an -dimensional simplex of the space (see [1]).

For problem 4) (see [2]), it has been conjectured that if is bounded, the inequality

holds.

Every illumination problem is closely linked to a special covering of the body (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)