|
|
(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> |
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) |
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) |