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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i0501401.png" /> be a convex body in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i0501402.png" />-dimensional linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i0501403.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i0501404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i0501405.png" /> be respectively its boundary and its interior, and assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i0501406.png" />. The best known illumination problems are the following.
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1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i0501407.png" /> be a certain direction in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i0501408.png" />. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i0501409.png" /> is called illuminated from the outside by the direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014010.png" /> if the straight line passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014011.png" /> parallel to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014012.png" /> passes through a certain point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014013.png" /> and if the direction of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014014.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014015.png" />. The minimum number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014016.png" /> of directions in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014017.png" /> is sought that is sufficient to illuminate the whole set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014018.png" />.
+
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2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014019.png" /> be a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014020.png" />. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014021.png" /> is called illuminated from the outside by the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014022.png" /> if the straight line defined by the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014024.png" /> passes through a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014025.png" /> and if the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014027.png" /> have the same direction. The minimum number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014028.png" /> of points from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014029.png" /> is sought that is sufficient to illuminate the whole set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014030.png" />.
+
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.
  
3) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014031.png" /> be a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014032.png" />. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014033.png" /> is illuminated from within by the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014034.png" /> if the straight line defined by the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014036.png" /> passes through a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014037.png" /> and if the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014039.png" /> have opposite directions. The minimum number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014040.png" /> of points from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014041.png" /> is sought that is sufficient to illuminate the whole set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014042.png" /> from within.
+
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 $.
  
4) A system of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014043.png" /> is said to be fixing for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014044.png" /> if it possesses the properties: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014045.png" /> is sufficient to illuminate the whole set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014046.png" /> from within; and b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014047.png" /> does not have any proper subset sufficient to illuminate the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014048.png" /> from within. The maximum number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014049.png" /> of points of a fixing system is sought for the body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014050.png" />.
+
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 $.
  
Problem 1) was proposed in connection with the [[Hadwiger hypothesis|Hadwiger hypothesis]] (see [[#References|[1]]]): The minimum number of bodies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014051.png" /> homothetic to a bounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014052.png" /> with homothety coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014054.png" />, sufficient for covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014055.png" />, satisfies the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014056.png" />, whereby the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014057.png" /> characterizes a parallelepiped. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014058.png" /> bounded, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014059.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014060.png" /> is unbounded, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014061.png" />, and there exist bodies such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014062.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014063.png" /> (see [[#References|[1]]]).
+
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.
  
Problem 2) was proposed in connection with problem 1). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014064.png" /> bounded, the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014065.png" /> holds. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014066.png" /> is not bounded, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014068.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014069.png" /> for any unbounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014070.png" /> takes one of the values 1, 2, 3, 4, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014071.png" /> (see [[#References|[1]]]).
+
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} $.
  
The solution of problem 3) takes the form: The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014072.png" /> is defined if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014073.png" /> is not a [[Cone|cone]]. In this case,
+
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]]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014074.png" /></td> </tr></table>
+
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]]]).
  
whereby <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014075.png" /> characterizes an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014076.png" />-dimensional simplex of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014077.png" /> (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,
  
For problem 4) (see [[#References|[2]]]), it has been conjectured that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014078.png" /> is bounded, the inequality
+
$$
 +
2 \leq  p( K) \leq  n+ 1 ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014079.png" /></td> </tr></table>
+
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.
 
holds.
  
Every illumination problem is closely linked to a special covering of the body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050140/i05014080.png" /> (cf. [[Covering (of a set)|Covering (of a set)]]) (see [[#References|[1]]]).
+
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====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 22:11, 5 June 2020


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

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)
How to Cite This Entry:
Illumination problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Illumination_problem&oldid=18579
This article was adapted from an original article by P.S. Soltan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article