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 |
Comments
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) |
Illumination problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Illumination_problem&oldid=18579