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A number characterizing the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v0961402.png" />-dimensional content of a set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v0961403.png" />-dimensional Euclidean space. The zero variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v0961404.png" /> of a closed bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v0961405.png" /> is the number of components of this set.
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In the simplest case of the plane, the linear variation of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v0961406.png" /> (i.e. the first-order variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v0961407.png" />) is the integral
+
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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/v/v096/v096140/v0961408.png" /></td> </tr></table>
+
A number characterizing the  $  k $-
 +
dimensional content of a set in  $  n $-
 +
dimensional Euclidean space. The zero variation  $  {V _ {0} } ( E) $
 +
of a closed bounded set  $  E $
 +
is the number of components of this set.
 +
 
 +
In the simplest case of the plane, the linear variation of a set  $  E $(
 +
i.e. the first-order variation of  $  E $)
 +
is the integral
 +
 
 +
$$
 +
V _ {1} ( E)  =  c \int\limits _ { 0 } ^ { {2 }  \pi } \Phi ( \alpha , E)  d \alpha
 +
$$
  
 
of the function
 
of the function
  
<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/v/v096/v096140/v0961409.png" /></td> </tr></table>
+
$$
 +
\Phi ( \alpha , E)  = \int\limits _ {\Pi _  \alpha  } V _ {0} ( E \cap \Pi _ {\alpha ,z }  ^  \perp  )  dz ,
 +
$$
  
where the integration is performed over the straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614010.png" /> passing through the coordinate origin, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614011.png" /> is the angle formed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614012.png" /> with a given axis and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614013.png" /> is the straight line normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614014.png" /> which intersects it at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614015.png" />. The normalizing constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614016.png" /> is so chosen that the variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614017.png" /> of an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614018.png" /> is equal to its length. For sufficiently simple sets, e.g. for rectifiable curves, the variation of the set is equal to the [[Length|length]] of the curve. For a closed domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614019.png" /> with a rectifiable boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614020.png" /> its linear variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614021.png" /> is equal to one-half the length of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614022.png" />. The second variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614023.png" /> (i.e. the second-order variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614024.png" />) is the two-dimensional measure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614025.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614026.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614027.png" />.
+
where the integration is performed over the straight line $  \Pi _  \alpha  $
 +
passing through the coordinate origin, $  \alpha $
 +
is the angle formed by $  \Pi _  \alpha  $
 +
with a given axis and $  \Pi _ {\alpha , z }  ^  \perp  $
 +
is the straight line normal to $  \Pi _  \alpha  $
 +
which intersects it at the point $  z $.  
 +
The normalizing constant $  c $
 +
is so chosen that the variation $  {V _ {1} } ( E) $
 +
of an interval $  E $
 +
is equal to its length. For sufficiently simple sets, e.g. for rectifiable curves, the variation of the set is equal to the [[Length|length]] of the curve. For a closed domain $  E $
 +
with a rectifiable boundary $  \Gamma $
 +
its linear variation $  {V _ {1} } ( E) $
 +
is equal to one-half the length of $  \Gamma $.  
 +
The second variation of $  E $(
 +
i.e. the second-order variation of $  E $)  
 +
is the two-dimensional measure of $  E $,  
 +
and $  {V _ {k} } ( E) = 0 $
 +
if $  k > 2 $.
  
In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614028.png" />-dimensional Euclidean space the variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614030.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614031.png" />, of a bounded closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614032.png" /> is the integral
+
In $  n $-
 +
dimensional Euclidean space the variation $  {V _ {i} } ( E) $
 +
of order 0 \dots n $,  
 +
of a bounded closed set $  E $
 +
is the integral
  
<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/v/v096/v096140/v09614033.png" /></td> </tr></table>
+
$$
 +
V _ {k} ( E)  = \
 +
\int\limits _ {\Omega _ {k}  ^ {n} }
 +
V _ {0} ( E \cap \beta )  d \mu _  \beta  $$
  
of the zero variation of the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614034.png" /> with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614035.png" />-dimensional plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614036.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614037.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614038.png" />-dimensional planes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614039.png" /> with respect to the [[Haar measure|Haar measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614040.png" />; normalized so that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614041.png" />-dimensional unit cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614042.png" /> has variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614043.png" />.
+
of the zero variation of the intersection of $  E $
 +
with an $  ( n - k) $-
 +
dimensional plane $  \beta $
 +
in the space $  \Omega _ {k}  ^ {n} $
 +
of all $  ( n - k) $-
 +
dimensional planes of $  \mathbf R  ^ {n} $
 +
with respect to the [[Haar measure|Haar measure]] $  d {\mu _  \beta  } $;  
 +
normalized so that the $  k $-
 +
dimensional unit cube $  J _ {k} $
 +
has variation $  {V _ {k} } ( J _ {k} ) = 1 $.
  
The variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614044.png" /> is identical with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614045.png" />-dimensional [[Lebesgue measure|Lebesgue measure]] of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614046.png" />. For convex bodies the (suitably normalized) set variations are identical with Minkowski's mixed volumes (cf. [[Mixed-volume theory|Mixed-volume theory]]) [[#References|[4]]].
+
The variation $  {V _ {n} } ( E) $
 +
is identical with the $  n $-
 +
dimensional [[Lebesgue measure|Lebesgue measure]] of the set $  E $.  
 +
For convex bodies the (suitably normalized) set variations are identical with Minkowski's mixed volumes (cf. [[Mixed-volume theory|Mixed-volume theory]]) [[#References|[4]]].
  
 
===Properties of the variations of a set.===
 
===Properties of the variations of a set.===
  
 
+
1) The variations $  {V _ {k} } ( E) $
1) The variations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614047.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614048.png" /> calculated for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614049.png" /> and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614050.png" /> have the same value.
+
for $  E \subset  \mathbf R  ^ {n} \subset  \mathbf R ^ {n  ^  \prime  } $
 +
calculated for $  E \subset  \mathbf R  ^ {n} $
 +
and for $  E \subset  \mathbf R ^ {n  ^  \prime  } $
 +
have the same value.
  
 
2) The variations of a set can be inductively expressed by the formula
 
2) The variations of a set can be inductively expressed 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/v/v096/v096140/v09614051.png" /></td> </tr></table>
+
$$
 +
\int\limits _ {\Omega _ {k}  ^ {n} }
 +
V _ {i} ( E \cap \beta )  d \mu _  \beta  = \
 +
c( n, k, i) V _ {k+} i ( E),\ \
 +
k+ i \leq  n ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614052.png" /> is the normalization constant.
+
where $  c ( n, k, i) $
 +
is the normalization constant.
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614053.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614054.png" />.
+
3) $  {V _ {i} } ( E) = 0 $
 +
implies $  {V _ {i+} 1 } ( E) = 0 $.
  
4) In a certain sense, the variations of a set are not dependent, i.e. for any sequence of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614055.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614056.png" /> is a positive integer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614057.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614058.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614059.png" />, it is possible to construct a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614060.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614062.png" />.
+
4) In a certain sense, the variations of a set are not dependent, i.e. for any sequence of numbers $  a _ {0} \dots a _ {n} $,  
 +
where $  a _ {0} $
 +
is a positive integer, $  0 < a _ {i} \leq  \infty $(
 +
$  i = 1 \dots n - 1 $),  
 +
$  a _ {n} = 0 $,  
 +
it is possible to construct a set $  E \subset  \mathbf R  ^ {n} $
 +
for which $  {V _ {i} } ( E) = a _ {i} $,  
 +
$  i = 0 \dots n $.
  
5) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614064.png" /> do not intersect, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614065.png" />. In the general case,
+
5) If $  E _ {1} $
 +
and $  E _ {2} $
 +
do not intersect, $  V _ {i} ( E _ {1} \cup E _ {2} ) = {V _ {i} } ( E _ {1} ) + {V _ {i} } ( E _ {2} ) $.  
 +
In the general case,
  
<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/v/v096/v096140/v09614066.png" /></td> </tr></table>
+
$$
 +
V _ {i} ( E _ {1} \cup E _ {2} )  \leq  \
 +
V _ {i} ( E _ {1} ) + V _ {i} ( E _ {2} ).
 +
$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614067.png" /> the variations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614068.png" /> are not monotone, i.e. it can happen for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614069.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614070.png" />.
+
For $  i = 0 \dots n - 1 $
 +
the variations $  V _ {i} $
 +
are not monotone, i.e. it can happen for $  E _ {1} \supset E _ {2} $
 +
that $  {V _ {i} } ( E _ {1} ) < {V _ {i} } ( E _ {2} ) $.
  
6) The variations of a set are semi-continuous, i.e. if a sequence of closed bounded sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614071.png" /> converges (in the sense of deviation in metric) to a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614072.png" />, then
+
6) The variations of a set are semi-continuous, i.e. if a sequence of closed bounded sets $  E _ {k} $
 +
converges (in the sense of deviation in metric) to a set $  E $,  
 +
then
  
<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/v/v096/v096140/v09614073.png" /></td> </tr></table>
+
$$
 +
V _ {0} ( E)  \leq  \
 +
{\lim\limits  \inf } _ {k \rightarrow \infty }
 +
V _ {0} ( E _ {n} ) ,
 +
$$
  
and if, in addition, the sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614074.png" /> are uniformly bounded, then
+
and if, in addition, the sums $  {V _ {0} } ( E _ {k} ) + \dots + {V _ {i-} 1 } ( E _ {k} ) $
 +
are uniformly bounded, then
  
<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/v/v096/v096140/v09614075.png" /></td> </tr></table>
+
$$
 +
V _ {i} ( E)  \leq  \
 +
{\lim\limits  \inf } _ {k \rightarrow \infty }  V _ {i} ( E _ {k} ) ,\ \
 +
i = 1 \dots n .
 +
$$
  
7) The variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614076.png" /> becomes identical with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614077.png" />-dimensional [[Hausdorff measure|Hausdorff measure]] if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096140/v09614078.png" /> and if
+
7) The variation $  {V _ {k} } ( E) $
 +
becomes identical with the $  k $-
 +
dimensional [[Hausdorff measure|Hausdorff measure]] if $  {V _ {k+} 1 } ( E) = 0 $
 +
and if
  
<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/v/v096/v096140/v09614079.png" /></td> </tr></table>
+
$$
 +
V _ {0} ( E) + \dots + V _ {k} ( E)  < \infty .
 +
$$
  
 
These conditions are met, for example, by twice-differentiable manifolds.
 
These conditions are met, for example, by twice-differentiable manifolds.
Line 58: Line 158:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Vitushkin,  "On higher-dimensional variations" , Moscow  (1955)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Vitushkin,  "Estimation of the complexity of the tabulation problem" , Moscow  (1959)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.G. Vitushkin,  "Proof of the upper semicontinuity of a set variation"  ''Soviet Math. Dokl.'' , '''7''' :  1  (1966)  pp. 206–209  ''Dokl. Akad. Nauk SSSR'' , '''166''' :  5  (1966)  pp. 1022–1025</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.M. Leontovich,  M.S. Mel'nikov,  "On the boundedness of the variations of a manifold"  ''Trans. Moscow Math Soc.'' , '''14'''  (1965)  pp. 333–368  ''Trudy Moskov. Mat. Obshch.'' , '''14'''  (1965)  pp. 306–337</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.D. Ivanov,  "Geometric properties of sets with finite variation"  ''Math. USSR-Sb.'' , '''1''' :  2  (1967)  pp. 405–427  ''Mat. Sb.'' , '''72 (114)''' :  3  (1967)  pp. 445–470</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L.D. Ivanov,  "On the local structure of sets with finite variation"  ''Math. USSR-Sb.'' , '''7''' :  1  (1969)  pp. 79–93  ''Mat. Sb.'' , '''78 (120)''' :  1  (1969)  pp. 85–100</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Vitushkin,  "On higher-dimensional variations" , Moscow  (1955)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.G. Vitushkin,  "Estimation of the complexity of the tabulation problem" , Moscow  (1959)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.G. Vitushkin,  "Proof of the upper semicontinuity of a set variation"  ''Soviet Math. Dokl.'' , '''7''' :  1  (1966)  pp. 206–209  ''Dokl. Akad. Nauk SSSR'' , '''166''' :  5  (1966)  pp. 1022–1025</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.M. Leontovich,  M.S. Mel'nikov,  "On the boundedness of the variations of a manifold"  ''Trans. Moscow Math Soc.'' , '''14'''  (1965)  pp. 333–368  ''Trudy Moskov. Mat. Obshch.'' , '''14'''  (1965)  pp. 306–337</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.D. Ivanov,  "Geometric properties of sets with finite variation"  ''Math. USSR-Sb.'' , '''1''' :  2  (1967)  pp. 405–427  ''Mat. Sb.'' , '''72 (114)''' :  3  (1967)  pp. 445–470</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L.D. Ivanov,  "On the local structure of sets with finite variation"  ''Math. USSR-Sb.'' , '''7''' :  1  (1969)  pp. 79–93  ''Mat. Sb.'' , '''78 (120)''' :  1  (1969)  pp. 85–100</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
Cf. also [[Content|Content]] and [[Variation of a function|Variation of a function]].
 
Cf. also [[Content|Content]] and [[Variation of a function|Variation of a function]].

Latest revision as of 08:27, 6 June 2020


A number characterizing the $ k $- dimensional content of a set in $ n $- dimensional Euclidean space. The zero variation $ {V _ {0} } ( E) $ of a closed bounded set $ E $ is the number of components of this set.

In the simplest case of the plane, the linear variation of a set $ E $( i.e. the first-order variation of $ E $) is the integral

$$ V _ {1} ( E) = c \int\limits _ { 0 } ^ { {2 } \pi } \Phi ( \alpha , E) d \alpha $$

of the function

$$ \Phi ( \alpha , E) = \int\limits _ {\Pi _ \alpha } V _ {0} ( E \cap \Pi _ {\alpha ,z } ^ \perp ) dz , $$

where the integration is performed over the straight line $ \Pi _ \alpha $ passing through the coordinate origin, $ \alpha $ is the angle formed by $ \Pi _ \alpha $ with a given axis and $ \Pi _ {\alpha , z } ^ \perp $ is the straight line normal to $ \Pi _ \alpha $ which intersects it at the point $ z $. The normalizing constant $ c $ is so chosen that the variation $ {V _ {1} } ( E) $ of an interval $ E $ is equal to its length. For sufficiently simple sets, e.g. for rectifiable curves, the variation of the set is equal to the length of the curve. For a closed domain $ E $ with a rectifiable boundary $ \Gamma $ its linear variation $ {V _ {1} } ( E) $ is equal to one-half the length of $ \Gamma $. The second variation of $ E $( i.e. the second-order variation of $ E $) is the two-dimensional measure of $ E $, and $ {V _ {k} } ( E) = 0 $ if $ k > 2 $.

In $ n $- dimensional Euclidean space the variation $ {V _ {i} } ( E) $ of order $ 0 \dots n $, of a bounded closed set $ E $ is the integral

$$ V _ {k} ( E) = \ \int\limits _ {\Omega _ {k} ^ {n} } V _ {0} ( E \cap \beta ) d \mu _ \beta $$

of the zero variation of the intersection of $ E $ with an $ ( n - k) $- dimensional plane $ \beta $ in the space $ \Omega _ {k} ^ {n} $ of all $ ( n - k) $- dimensional planes of $ \mathbf R ^ {n} $ with respect to the Haar measure $ d {\mu _ \beta } $; normalized so that the $ k $- dimensional unit cube $ J _ {k} $ has variation $ {V _ {k} } ( J _ {k} ) = 1 $.

The variation $ {V _ {n} } ( E) $ is identical with the $ n $- dimensional Lebesgue measure of the set $ E $. For convex bodies the (suitably normalized) set variations are identical with Minkowski's mixed volumes (cf. Mixed-volume theory) [4].

Properties of the variations of a set.

1) The variations $ {V _ {k} } ( E) $ for $ E \subset \mathbf R ^ {n} \subset \mathbf R ^ {n ^ \prime } $ calculated for $ E \subset \mathbf R ^ {n} $ and for $ E \subset \mathbf R ^ {n ^ \prime } $ have the same value.

2) The variations of a set can be inductively expressed by the formula

$$ \int\limits _ {\Omega _ {k} ^ {n} } V _ {i} ( E \cap \beta ) d \mu _ \beta = \ c( n, k, i) V _ {k+} i ( E),\ \ k+ i \leq n , $$

where $ c ( n, k, i) $ is the normalization constant.

3) $ {V _ {i} } ( E) = 0 $ implies $ {V _ {i+} 1 } ( E) = 0 $.

4) In a certain sense, the variations of a set are not dependent, i.e. for any sequence of numbers $ a _ {0} \dots a _ {n} $, where $ a _ {0} $ is a positive integer, $ 0 < a _ {i} \leq \infty $( $ i = 1 \dots n - 1 $), $ a _ {n} = 0 $, it is possible to construct a set $ E \subset \mathbf R ^ {n} $ for which $ {V _ {i} } ( E) = a _ {i} $, $ i = 0 \dots n $.

5) If $ E _ {1} $ and $ E _ {2} $ do not intersect, $ V _ {i} ( E _ {1} \cup E _ {2} ) = {V _ {i} } ( E _ {1} ) + {V _ {i} } ( E _ {2} ) $. In the general case,

$$ V _ {i} ( E _ {1} \cup E _ {2} ) \leq \ V _ {i} ( E _ {1} ) + V _ {i} ( E _ {2} ). $$

For $ i = 0 \dots n - 1 $ the variations $ V _ {i} $ are not monotone, i.e. it can happen for $ E _ {1} \supset E _ {2} $ that $ {V _ {i} } ( E _ {1} ) < {V _ {i} } ( E _ {2} ) $.

6) The variations of a set are semi-continuous, i.e. if a sequence of closed bounded sets $ E _ {k} $ converges (in the sense of deviation in metric) to a set $ E $, then

$$ V _ {0} ( E) \leq \ {\lim\limits \inf } _ {k \rightarrow \infty } V _ {0} ( E _ {n} ) , $$

and if, in addition, the sums $ {V _ {0} } ( E _ {k} ) + \dots + {V _ {i-} 1 } ( E _ {k} ) $ are uniformly bounded, then

$$ V _ {i} ( E) \leq \ {\lim\limits \inf } _ {k \rightarrow \infty } V _ {i} ( E _ {k} ) ,\ \ i = 1 \dots n . $$

7) The variation $ {V _ {k} } ( E) $ becomes identical with the $ k $- dimensional Hausdorff measure if $ {V _ {k+} 1 } ( E) = 0 $ and if

$$ V _ {0} ( E) + \dots + V _ {k} ( E) < \infty . $$

These conditions are met, for example, by twice-differentiable manifolds.

The concept of the variation of a set arose in the context of solutions of the Cauchy–Riemann system, and its ultimate formulation is due to A.G. Vitushkin. The set variations proved to be a useful tool in solving certain problems in analysis, in particular that of superposition of functions of several variables [1], and also in approximation problems [2].

References

[1] A.G. Vitushkin, "On higher-dimensional variations" , Moscow (1955) (In Russian)
[2] A.G. Vitushkin, "Estimation of the complexity of the tabulation problem" , Moscow (1959) (In Russian)
[3] A.G. Vitushkin, "Proof of the upper semicontinuity of a set variation" Soviet Math. Dokl. , 7 : 1 (1966) pp. 206–209 Dokl. Akad. Nauk SSSR , 166 : 5 (1966) pp. 1022–1025
[4] A.M. Leontovich, M.S. Mel'nikov, "On the boundedness of the variations of a manifold" Trans. Moscow Math Soc. , 14 (1965) pp. 333–368 Trudy Moskov. Mat. Obshch. , 14 (1965) pp. 306–337
[5] L.D. Ivanov, "Geometric properties of sets with finite variation" Math. USSR-Sb. , 1 : 2 (1967) pp. 405–427 Mat. Sb. , 72 (114) : 3 (1967) pp. 445–470
[6] L.D. Ivanov, "On the local structure of sets with finite variation" Math. USSR-Sb. , 7 : 1 (1969) pp. 79–93 Mat. Sb. , 78 (120) : 1 (1969) pp. 85–100

Comments

Cf. also Content and Variation of a function.

How to Cite This Entry:
Variation of a set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Variation_of_a_set&oldid=49116
This article was adapted from an original article by A.G. VitushkinL.D. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article