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A real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r0777001.png" /> defined on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r0777002.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r0777003.png" /> and satisfying the following conditions: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r0777004.png" /> is continuous, non-negative and homogeneous (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r0777005.png" /> for any real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r0777006.png" />). A ray function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r0777007.png" /> is said to be positive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r0777008.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r0777009.png" />, and symmetric if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770010.png" />. A ray function is said to be convex if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770011.png" />,
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<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/r/r077/r077700/r07770012.png" /></td> </tr></table>
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For any ray function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770013.png" /> there is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770014.png" /> for which
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A real-valued function $  F ( x) $
 +
defined on an  $  n $-
 +
dimensional space  $  \mathbf R  ^ {n} $
 +
and satisfying the following conditions:  $  F ( x) $
 +
is continuous, non-negative and homogeneous (that is,  $  F ( \tau x) = \tau F ( x) $
 +
for any real number  $  \tau \geq  0 $).  
 +
A ray function  $  F ( x) $
 +
is said to be positive if  $  F ( x) > 0 $
 +
for all  $  x \neq 0 $,
 +
and symmetric if  $  F ( - x ) = F ( x) $.  
 +
A ray function is said to be convex if for any  $  x , y \in \mathbf R  ^ {n} $,
  
<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/r/r077/r077700/r07770015.png" /></td> </tr></table>
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$$
 +
F ( x + y )  \leq  F ( x) + F ( y) .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770016.png" /> is positive, then there is also a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770017.png" /> for which
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For any ray function  $  F ( x) $
 +
there is a constant $  c = c _ {F} $
 +
for which
  
<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/r/r077/r077700/r07770018.png" /></td> </tr></table>
+
$$
 +
F ( x)  \leq  c  | x | ,\  x \in \mathbf R  ^ {n} .
 +
$$
  
The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770019.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770020.png" /> satisfying the condition
+
If  $  F ( x) $
 +
is positive, then there is also a constant  $  \widetilde{c}  = \widetilde{c}  _ {F} > 0 $
 +
for which
  
<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/r/r077/r077700/r07770021.png" /></td> </tr></table>
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$$
 +
F ( x)  \geq  \widetilde{c}  | x | ,\  x \in \mathbf R  ^ {n} .
 +
$$
  
is a [[Star body|star body]]. Conversely, for any open star body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770022.png" /> there is a unique ray function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770023.png" /> for which
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The set  $  \mathfrak C $
 +
of points  $  x \in \mathbf R  ^ {n} $
 +
satisfying the condition
  
<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/r/r077/r077700/r07770024.png" /></td> </tr></table>
+
$$
 +
F ( x)  < 1
 +
$$
  
A star body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770025.png" /> is bounded if and only if its ray function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770026.png" /> is positive. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770027.png" /> is a symmetric function, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770028.png" /> is symmetric about the point 0; the converse is also true. A star body is convex if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077700/r07770029.png" /> is a convex ray function.
+
is a [[Star body|star body]]. Conversely, for any open star body $  \mathfrak C $
 +
there is a unique ray function  $  F _ {\mathfrak C }  ( x) $
 +
for which
 +
 
 +
$$
 +
\mathfrak C  = \{ {x } : {F _ {\mathfrak C }  ( x) < 1 } \}
 +
.
 +
$$
 +
 
 +
A star body  $  \mathfrak C _ {F} $
 +
is bounded if and only if its ray function $  F ( x) $
 +
is positive. If $  F ( x) $
 +
is a symmetric function, then $  \mathfrak C _ {F} $
 +
is symmetric about the point 0; the converse is also true. A star body is convex if and only if $  F ( x) $
 +
is a convex ray function.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.W.S. Cassels,  "An introduction to the geometry of numbers" , Springer  (1959)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.W.S. Cassels,  "An introduction to the geometry of numbers" , Springer  (1959)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:10, 6 June 2020


A real-valued function $ F ( x) $ defined on an $ n $- dimensional space $ \mathbf R ^ {n} $ and satisfying the following conditions: $ F ( x) $ is continuous, non-negative and homogeneous (that is, $ F ( \tau x) = \tau F ( x) $ for any real number $ \tau \geq 0 $). A ray function $ F ( x) $ is said to be positive if $ F ( x) > 0 $ for all $ x \neq 0 $, and symmetric if $ F ( - x ) = F ( x) $. A ray function is said to be convex if for any $ x , y \in \mathbf R ^ {n} $,

$$ F ( x + y ) \leq F ( x) + F ( y) . $$

For any ray function $ F ( x) $ there is a constant $ c = c _ {F} $ for which

$$ F ( x) \leq c | x | ,\ x \in \mathbf R ^ {n} . $$

If $ F ( x) $ is positive, then there is also a constant $ \widetilde{c} = \widetilde{c} _ {F} > 0 $ for which

$$ F ( x) \geq \widetilde{c} | x | ,\ x \in \mathbf R ^ {n} . $$

The set $ \mathfrak C $ of points $ x \in \mathbf R ^ {n} $ satisfying the condition

$$ F ( x) < 1 $$

is a star body. Conversely, for any open star body $ \mathfrak C $ there is a unique ray function $ F _ {\mathfrak C } ( x) $ for which

$$ \mathfrak C = \{ {x } : {F _ {\mathfrak C } ( x) < 1 } \} . $$

A star body $ \mathfrak C _ {F} $ is bounded if and only if its ray function $ F ( x) $ is positive. If $ F ( x) $ is a symmetric function, then $ \mathfrak C _ {F} $ is symmetric about the point 0; the converse is also true. A star body is convex if and only if $ F ( x) $ is a convex ray function.

References

[1] J.W.S. Cassels, "An introduction to the geometry of numbers" , Springer (1959)

Comments

Star bodies are usually defined as closed ray sets. A ray function is more commonly called a distance function.

References

[a1] P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) pp. Sect. (iv) (Updated reprint)
[a2] E. Hlawka, "Das inhomogene Problem in der Geometrie der Zahlen" , Proc. Internat. Congress Mathematicians (Amsterdam, 1954) , 3 , Noordhoff (1954) pp. 20–27 ((Also: Selecta, Springer 1990, 178–185.))
How to Cite This Entry:
Ray function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ray_function&oldid=48442
This article was adapted from an original article by A.V. Malyshev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article