View source for Illumination problem

<!--
i0501401.png
$#A+1 = 80 n = 0
$#C+1 = 80 : ~/encyclopedia/old_files/data/I050/I.0500140 Illumination problem
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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|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|Hadwiger hypothesis]] (see [[#References|[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 [[#References|[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 [[#References|[1]]]).

The solution of problem 3) takes the form: The number  $  p( K) $
is defined if and only if  $  K $
is not a [[Cone|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 [[#References|[1]]]).

For problem 4) (see [[#References|[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)|Covering (of a set)]]) (see [[#References|[1]]]).

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

====Comments====

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