Namespaces
Variants
Actions

Difference between revisions of "Star body"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(latex details)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
''with respect to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s0872001.png" />, star-like body''
+
<!--
 +
s0872001.png
 +
$#A+1 = 50 n = 0
 +
$#C+1 = 50 : ~/encyclopedia/old_files/data/S087/S.0807200 Star body
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
An open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s0872002.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s0872003.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s0872004.png" /> which has the ray property (relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s0872005.png" />): If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s0872006.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s0872007.png" /> is the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s0872008.png" />, then the entire segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s0872009.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720011.png" />) lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720012.png" />. A star body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720013.png" /> with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720014.png" /> may be characterized as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720015.png" /> is an interior point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720016.png" />; every ray emanating from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720017.png" /> lies either entirely in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720018.png" /> or contains a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720019.png" /> such that the ray segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720020.png" /> lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720021.png" />, but the ray segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720022.png" /> lies outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720023.png" />. This definition is equivalent to the first one, up to points on the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720024.png" />. A star body is a particular case of a star set with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720025.png" />, a set with the generalized ray property relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720026.png" />: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720027.png" />, then the entire segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720028.png" /> lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720029.png" />. A particular case of a star body is a [[Convex body|convex body]].
+
{{TEX|auto}}
 +
{{TEX|done}}
  
With every star body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720030.png" /> with respect to the origin <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720031.png" /> one can associate, in one-to-one fashion, a [[Ray function|ray function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720032.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720033.png" /> is the set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720034.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720035.png" />.
+
''with respect to a point  $  O $,
 +
star-like body''
 +
 
 +
An open set  $  \mathfrak S $
 +
in  $  n $-
 +
dimensional Euclidean space  $  \mathbf R  ^ {n} $
 +
which has the ray property (relative to  $  O $):  
 +
If  $  a \in \overline{\mathfrak S}\; $,
 +
where  $  \overline{\mathfrak S}\; $
 +
is the closure of  $  \mathfrak S $,
 +
then the entire segment  $  [ O , a ) $(
 +
where  $  O \in [ O , a ) $,
 +
$  a \notin [ O , a ) $)
 +
lies in  $  \mathfrak S $.  
 +
A star body  $  \mathfrak S $
 +
with centre  $  O $
 +
may be characterized as follows:  $  O $
 +
is an interior point of  $  \mathfrak S $;
 +
every ray emanating from  $  O $
 +
lies either entirely in  $  \mathfrak S $
 +
or contains a point  $  a $
 +
such that the ray segment  $  [ O , a ) $
 +
lies in  $  \mathfrak S $,
 +
but the ray segment  $  ( a, + \infty ) $
 +
lies outside  $  \mathfrak S $.  
 +
This definition is equivalent to the first one, up to points on the boundary of  $  \mathfrak S $.  
 +
A star body is a particular case of a star set with respect to $  O $,
 +
a set with the generalized ray property relative to  $  O $:  
 +
If  $  a \in \mathfrak S $,
 +
then the entire segment  $  [ O , a ] $
 +
lies in  $  \mathfrak S $.  
 +
A particular case of a star body is a [[Convex body|convex body]].
 +
 
 +
With every star body  $  \mathfrak S $
 +
with respect to the origin  $  O $
 +
one can associate, in one-to-one fashion, a [[Ray function|ray function]] $  F ( x ) = F _ {\mathfrak S} ( x ) $
 +
such that $  \mathfrak S $
 +
is the set of points $  x \in \mathbf R  ^ {n} $
 +
with $  F ( x ) < 1 $.
  
 
The correspondence is defined by the formula
 
The correspondence is defined by the formula
  
<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/s/s087/s087200/s08720036.png" /></td> </tr></table>
+
$$
 +
F ( x )  = \inf _ {\begin{array}{c}
 +
{tx \in \mathfrak S } \\
 +
{t > 0 }
 +
\end{array}
 +
}
 +
 +
\frac{1}{t}
 +
.
 +
$$
  
With this notation a star body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720037.png" /> is bounded if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720038.png" /> is a positive ray function; it is convex if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720039.png" /> is a convex ray function.
+
With this notation a star body $  \mathfrak S $
 +
is bounded if and only if $  F ( x ) $
 +
is a positive ray function; it 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  (1972)</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  (1972)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Star bodies play an important role in the [[Geometry of numbers|geometry of numbers]], e.g. the Minkowski–Hlawka theorem.
+
Star bodies play an important role in the [[geometry of numbers]], ''e.g.'' the Minkowski–Hlawka theorem.
  
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720040.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720041.png" /> is centrally symmetric if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720042.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720043.png" />.
+
A set $S$ in $  \mathbf R  ^ {n} $
 +
is centrally symmetric if $  x \in S $
 +
implies $  - x \in S $.
  
The Minkowski–Hlawka theorem says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720044.png" /> for a centrally-symmetric star body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720045.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720046.png" /> is the critical determinant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720047.png" /> (cf. [[Geometry of numbers|Geometry of numbers]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720048.png" /> is the volume of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087200/s08720050.png" />. This is an inequality in the opposite direction of the Minkowski convex body theorem (cf. [[Minkowski theorem|Minkowski theorem]]).
+
The Minkowski–Hlawka theorem says that $  V ( \mathfrak S ) \geq  2 \zeta ( n) \Delta ( \mathfrak S ) $
 +
for a centrally-symmetric star body $  \mathfrak S $.  
 +
Here, $  \Delta ( \mathfrak S ) $
 +
is the critical determinant of $  \mathfrak S $(
 +
cf. [[Geometry of numbers]]), $  V( \mathfrak S ) $
 +
is the volume of $  \mathfrak S $
 +
and $\zeta(n) = 1+ 2^{-n} + 3^{-n} + \dots $.  
 +
This is an inequality in the opposite direction of the Minkowski convex body theorem (cf. [[Minkowski theorem]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Gruber,   C.G. Lekkerkerker,   "Geometry of numbers" , North-Holland  (1987)  pp. Sect. (iv)  (Updated reprint)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Erdös,   P.M. Gruber,   J. Hammer,   "Lattice points" , Longman  (1989)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland  (1987)  pp. Sect. (iv)  (Updated reprint)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Erdös, P.M. Gruber, J. Hammer, "Lattice points" , Longman  (1989)</TD></TR>
 +
</table>

Latest revision as of 20:42, 16 January 2024


with respect to a point $ O $, star-like body

An open set $ \mathfrak S $ in $ n $- dimensional Euclidean space $ \mathbf R ^ {n} $ which has the ray property (relative to $ O $): If $ a \in \overline{\mathfrak S}\; $, where $ \overline{\mathfrak S}\; $ is the closure of $ \mathfrak S $, then the entire segment $ [ O , a ) $( where $ O \in [ O , a ) $, $ a \notin [ O , a ) $) lies in $ \mathfrak S $. A star body $ \mathfrak S $ with centre $ O $ may be characterized as follows: $ O $ is an interior point of $ \mathfrak S $; every ray emanating from $ O $ lies either entirely in $ \mathfrak S $ or contains a point $ a $ such that the ray segment $ [ O , a ) $ lies in $ \mathfrak S $, but the ray segment $ ( a, + \infty ) $ lies outside $ \mathfrak S $. This definition is equivalent to the first one, up to points on the boundary of $ \mathfrak S $. A star body is a particular case of a star set with respect to $ O $, a set with the generalized ray property relative to $ O $: If $ a \in \mathfrak S $, then the entire segment $ [ O , a ] $ lies in $ \mathfrak S $. A particular case of a star body is a convex body.

With every star body $ \mathfrak S $ with respect to the origin $ O $ one can associate, in one-to-one fashion, a ray function $ F ( x ) = F _ {\mathfrak S} ( x ) $ such that $ \mathfrak S $ is the set of points $ x \in \mathbf R ^ {n} $ with $ F ( x ) < 1 $.

The correspondence is defined by the formula

$$ F ( x ) = \inf _ {\begin{array}{c} {tx \in \mathfrak S } \\ {t > 0 } \end{array} } \frac{1}{t} . $$

With this notation a star body $ \mathfrak S $ is bounded if and only if $ F ( x ) $ is a positive ray function; it 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 (1972)

Comments

Star bodies play an important role in the geometry of numbers, e.g. the Minkowski–Hlawka theorem.

A set $S$ in $ \mathbf R ^ {n} $ is centrally symmetric if $ x \in S $ implies $ - x \in S $.

The Minkowski–Hlawka theorem says that $ V ( \mathfrak S ) \geq 2 \zeta ( n) \Delta ( \mathfrak S ) $ for a centrally-symmetric star body $ \mathfrak S $. Here, $ \Delta ( \mathfrak S ) $ is the critical determinant of $ \mathfrak S $( cf. Geometry of numbers), $ V( \mathfrak S ) $ is the volume of $ \mathfrak S $ and $\zeta(n) = 1+ 2^{-n} + 3^{-n} + \dots $. This is an inequality in the opposite direction of the Minkowski convex body theorem (cf. Minkowski theorem).

References

[a1] P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) pp. Sect. (iv) (Updated reprint)
[a2] P. Erdös, P.M. Gruber, J. Hammer, "Lattice points" , Longman (1989)
How to Cite This Entry:
Star body. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Star_body&oldid=11275
This article was adapted from an original article by A.V. Malyshev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article