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''in the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s0848202.png" />-dimensional polyhedral chains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s0848203.png" />''
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$#A+1 = 86 n = 0
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$#C+1 = 86 : ~/encyclopedia/old_files/data/S084/S.0804820 Sharp norm
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The largest semi-norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s0848204.png" /> which, for any cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s0848205.png" /> of volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s0848206.png" />, satisfies the inequalities
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/s084/s084820/s0848207.png" /></td> </tr></table>
+
''in the space of  $  r $-
 +
dimensional polyhedral chains  $  C _ {r} ( E  ^ {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/s/s084/s084820/s0848208.png" /></td> </tr></table>
+
The largest semi-norm  $  {| \cdot | }  ^  \prime  $
 +
which, for any cell  $  \sigma  ^ {r} $
 +
of volume  $  | \sigma  ^ {r} | $,
 +
satisfies the inequalities
  
<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/s084/s084820/s0848209.png" /></td> </tr></table>
+
$$
 +
{| \sigma  ^ {r} | }  ^  \prime  \leq  | \sigma  ^ {r} | ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482010.png" /> is the cell obtained by shifting by a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482011.png" /> of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482012.png" />.
+
$$
 +
{| \partial  \sigma  ^ {r+} 1 | }  ^  \prime  \leq  | \sigma  ^ {r+} 1 | ,
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482013.png" />, the sharp norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482014.png" /> is expressed as follows:
+
$$
 +
{| T _ {v} \sigma  ^ {r} - \sigma  ^ {r} | }  ^  \prime  \leq 
 +
\frac{
 +
{| \sigma  ^ {r} | }  ^  \prime  | v | }{r+}
 +
1 ,
 +
$$
  
<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/s/s084/s084820/s08482015.png" /></td> </tr></table>
+
where  $  T _ {v} \sigma  ^ {r} $
 +
is the cell obtained by shifting by a vector  $  v $
 +
of length  $  | v | $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482016.png" /> is the [[Flat norm|flat norm]] of the chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482017.png" />, and the infimum is taken over all shifts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482018.png" />.
+
If  $  A = \sum a _ {i} \sigma _ {i}  ^ {r} $,
 +
the sharp norm  $  | A |  ^  \srp  $
 +
is expressed as follows:
 +
 
 +
$$
 +
| A |  ^  \srp  = \inf \left \{
 +
\frac{\sum | a _ {i} |  | \sigma _ {i}  ^ {r} |  | v _ {i} | }{r+}
 +
1 + \left |
 +
\sum a _ {i} T _ {v _ {i}  } \sigma _ {i}  ^ {r} \right |  ^  \flt  \right \} ,
 +
$$
 +
 
 +
where  $  | C |  ^  \flt  $
 +
is the [[Flat norm|flat norm]] of the chain $  C $,  
 +
and the infimum is taken over all shifts $  v $.
  
 
One has
 
One has
  
<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/s084/s084820/s08482019.png" /></td> </tr></table>
+
$$
 +
| a A |  ^  \srp  = | a |  | A |  ^  \srp  ,
 +
$$
  
<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/s084/s084820/s08482020.png" /></td> </tr></table>
+
$$
 +
| A + B |  ^  \srp  \leq  | A |  ^  \srp  + | B |  ^  \srp  ,
 +
$$
  
<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/s084/s084820/s08482021.png" /></td> </tr></table>
+
$$
 +
| A |  ^  \srp  = 0  \iff  A = 0 ,
 +
$$
  
<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/s084/s084820/s08482022.png" /></td> </tr></table>
+
$$
 +
| A |  ^  \srp  \leq  | A |  ^  \flt  ;
 +
$$
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482024.png" />.
+
if $  r = 0 $,  
 +
then $  | A |  ^  \srp  = | A |  ^  \flt  $.
  
The completion of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482025.png" /> is the separable Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482026.png" />, whose elements are known as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482027.png" />-dimensional sharp chains. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482028.png" />-dimensional polyhedral chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482029.png" /> and any vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482030.png" />,
+
The completion of the space $  C _ {r} ( E  ^ {n} ) $
 +
is the separable Banach space $  C _ {r}  ^  \srp  ( E  ^ {n} ) $,  
 +
whose elements are known as $  r $-
 +
dimensional sharp chains. For any $  r $-
 +
dimensional polyhedral chain $  A $
 +
and any vector $  v $,
  
<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/s084/s084820/s08482031.png" /></td> </tr></table>
+
$$
 +
| T _ {v} A - A |  ^  \srp  \leq 
 +
\frac{| A |  | v | }{r+}
 +
1 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482032.png" /> is the chain obtained by shifting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482033.png" /> by the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482034.png" /> of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482035.png" />. A flat chain of finite mass is a sharp chain; in general, any flat chain may also be considered as a sharp chain in the following sense: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482037.png" /> are polyhedral chains, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482038.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482039.png" /> is a linear bijective mapping from the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482040.png" /> into the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482041.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482042.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482043.png" /> in the sharp norm.
+
where $  T _ {v} A $
 +
is the chain obtained by shifting $  A $
 +
by the vector $  v $
 +
of length $  | v | $.  
 +
A flat chain of finite mass is a sharp chain; in general, any flat chain may also be considered as a sharp chain in the following sense: If $  A = \lim\limits  ^  \flt  A _ {i} $,  
 +
where $  A _ {i} $
 +
are polyhedral chains, and $  \psi A = \lim\limits  ^  \srp  A _ {i} $,  
 +
where $  \psi $
 +
is a linear bijective mapping from the space $  C _ {r}  ^  \flt  ( E  ^ {n)} $
 +
into the space $  C _ {r}  ^  \srp  ( E  ^ {n)} $,  
 +
and $  \psi C _ {r}  ^  \flt  $
 +
is dense in $  C _ {r}  ^  \srp  $
 +
in the sharp norm.
  
It is not possible to give a correct definition of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482044.png" /> of a sharp chain [[#References|[1]]]; an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482045.png" />-dimensional sharp chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482046.png" /> is an element of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482047.png" /> dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482048.png" />; it is a flat cochain, and
+
It is not possible to give a correct definition of the boundary $  \partial  A $
 +
of a sharp chain [[#References|[1]]]; an $  r $-
 +
dimensional sharp chain $  X = XA $
 +
is an element of the space $  C ^ {\srp r } ( E  ^ {n} ) $
 +
dual to $  C _ {r}  ^  \srp  ( E  ^ {n} ) $;  
 +
it is a flat cochain, and
  
<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/s084/s084820/s08482049.png" /></td> </tr></table>
+
$$
 +
| X |  \leq  | X |  ^  \flt  \leq  | X |  ^  \srp  ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482050.png" /> is the co-mass of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482051.png" />, while the sharp co-norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482052.png" /> is defined similarly to the flat norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482053.png" />. The co-boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482054.png" /> of a sharp cochain is not necessarily sharp [[#References|[1]]], but
+
where $  | X | $
 +
is the co-mass of $  X $,  
 +
while the sharp co-norm $  | X |  ^  \srp  $
 +
is defined similarly to the flat norm $  | X |  ^  \flt  $.  
 +
The co-boundary $  dX $
 +
of a sharp cochain is not necessarily sharp [[#References|[1]]], but
  
<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/s084/s084820/s08482055.png" /></td> </tr></table>
+
$$
 +
| dX |  \leq  | X |  ^  \flt  \leq  | X |  ^  \srp  .
 +
$$
  
The Lipschitz constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482056.png" /> of a cochain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482057.png" /> is defined as follows:
+
The Lipschitz constant $  {\mathcal L} ( X) $
 +
of a cochain $  X $
 +
is defined as follows:
  
<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/s084/s084820/s08482058.png" /></td> </tr></table>
+
$$
 +
{\mathcal L} ( X)  = \sup \left \{
 +
\frac{| X \cdot ( T _ {v} A - A ) |
 +
}{| A  | | v | }
 +
\right \} ,
 +
$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482059.png" /> are polyhedral chains. For sharp cochains this supremum is finite, and
+
where the $  A $
 +
are polyhedral chains. For sharp cochains this supremum is finite, and
  
<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/s084/s084820/s08482060.png" /></td> </tr></table>
+
$$
 +
( r + 1 ) {\mathcal L} ( X)  \leq  | X |  ^  \srp  .
 +
$$
  
 
Any flat cochain with a finite Lipschitz constant is sharp, and
 
Any flat cochain with a finite Lipschitz constant is sharp, and
  
<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/s084/s084820/s08482061.png" /></td> </tr></table>
+
$$
 +
| X |  ^  \srp  = \sup \{ | X |  ^  \flt  ,\
 +
( r + 1 ) {\mathcal L} ( X) \} ,
 +
$$
  
 
and also
 
and also
  
<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/s084/s084820/s08482062.png" /></td> </tr></table>
+
$$
 +
| dX |  \leq  ( r + 1 ) {\mathcal L} ( X) .
 +
$$
 +
 
 +
Similar concepts are introduced for  $  r $-
 +
dimensional polyhedral chains in open subsets  $  R \subset  E  ^ {n} $.  
 +
See also [[Sharp form|Sharp form]].
  
Similar concepts are introduced for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482063.png" />-dimensional polyhedral chains in open subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482064.png" />. See also [[Sharp form|Sharp form]].
+
The sharp norm in the space of additive functions  $  \gamma $
 +
whose values are $  r $-
 +
vectors is the largest of the semi-norms  $  | \cdot |  ^  \prime  $
 +
which satisfy the conditions:
  
The sharp norm in the space of additive functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482065.png" /> whose values are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482067.png" />-vectors is the largest of the semi-norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482068.png" /> which satisfy the conditions:
+
$  | \gamma |  ^  \prime  \leq  | \gamma | $,
 +
where  $  | \gamma | $
 +
is the complete variation of $  \gamma $;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482069.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482070.png" /> is the complete variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482071.png" />;
+
$$
 +
| T _ {v} \gamma - \gamma |  ^  \prime  \leq  \
  
<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/s084/s084820/s08482072.png" /></td> </tr></table>
+
\frac{| v |  | \gamma | }{r+}
 +
1 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482073.png" /> is the shift of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482074.png" /> by the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482075.png" /> of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482076.png" />:
+
where $  T _ {v} \gamma ( Q) = \gamma T _ {-} v ( Q) $
 +
is the shift of the function $  \gamma $
 +
by the vector $  v $
 +
of length $  | v | $:
  
<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/s084/s084820/s08482077.png" /></td> </tr></table>
+
$$
 +
T _ {-} v ( Q)  = \{ {q - v } : {q \in Q \subset  E  ^ {n} } \}
 +
;
 +
$$
  
for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482078.png" /> and an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482079.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482080.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482081.png" /> if the support <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482083.png" />.
+
for each point $  p $
 +
and an arbitrary $  \epsilon $
 +
there exists an $  \eta > 0 $
 +
such that $  | \gamma |  ^  \prime  \leq  \epsilon | \gamma | $
 +
if the support $  \supp  t \gamma \subset  U _  \eta  ( p) $
 +
and $  \gamma ( E  ^ {n)} = 0 $.
  
The sharp norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482084.png" /> is represented as follows:
+
The sharp norm $  | \gamma |  ^  \srp  $
 +
is represented as follows:
  
<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/s084/s084820/s08482085.png" /></td> </tr></table>
+
$$
 +
| \gamma |  ^  \srp  = \sup _  \omega  \int\limits _ {E  ^ {n}
 +
} \omega  d \gamma ,
 +
$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482086.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482087.png" />-dimensional sharp forms for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084820/s08482088.png" />.
+
where the $  \omega $
 +
are $  r $-
 +
dimensional sharp forms for which $  | \omega |  ^  \srp  \leq  1 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Whitney,  "Geometric integration theory" , Princeton Univ. Press  (1957)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Whitney,  "Geometric integration theory" , Princeton Univ. Press  (1957)</TD></TR></table>

Revision as of 08:13, 6 June 2020


in the space of $ r $- dimensional polyhedral chains $ C _ {r} ( E ^ {n)} $

The largest semi-norm $ {| \cdot | } ^ \prime $ which, for any cell $ \sigma ^ {r} $ of volume $ | \sigma ^ {r} | $, satisfies the inequalities

$$ {| \sigma ^ {r} | } ^ \prime \leq | \sigma ^ {r} | , $$

$$ {| \partial \sigma ^ {r+} 1 | } ^ \prime \leq | \sigma ^ {r+} 1 | , $$

$$ {| T _ {v} \sigma ^ {r} - \sigma ^ {r} | } ^ \prime \leq \frac{ {| \sigma ^ {r} | } ^ \prime | v | }{r+} 1 , $$

where $ T _ {v} \sigma ^ {r} $ is the cell obtained by shifting by a vector $ v $ of length $ | v | $.

If $ A = \sum a _ {i} \sigma _ {i} ^ {r} $, the sharp norm $ | A | ^ \srp $ is expressed as follows:

$$ | A | ^ \srp = \inf \left \{ \frac{\sum | a _ {i} | | \sigma _ {i} ^ {r} | | v _ {i} | }{r+} 1 + \left | \sum a _ {i} T _ {v _ {i} } \sigma _ {i} ^ {r} \right | ^ \flt \right \} , $$

where $ | C | ^ \flt $ is the flat norm of the chain $ C $, and the infimum is taken over all shifts $ v $.

One has

$$ | a A | ^ \srp = | a | | A | ^ \srp , $$

$$ | A + B | ^ \srp \leq | A | ^ \srp + | B | ^ \srp , $$

$$ | A | ^ \srp = 0 \iff A = 0 , $$

$$ | A | ^ \srp \leq | A | ^ \flt ; $$

if $ r = 0 $, then $ | A | ^ \srp = | A | ^ \flt $.

The completion of the space $ C _ {r} ( E ^ {n} ) $ is the separable Banach space $ C _ {r} ^ \srp ( E ^ {n} ) $, whose elements are known as $ r $- dimensional sharp chains. For any $ r $- dimensional polyhedral chain $ A $ and any vector $ v $,

$$ | T _ {v} A - A | ^ \srp \leq \frac{| A | | v | }{r+} 1 , $$

where $ T _ {v} A $ is the chain obtained by shifting $ A $ by the vector $ v $ of length $ | v | $. A flat chain of finite mass is a sharp chain; in general, any flat chain may also be considered as a sharp chain in the following sense: If $ A = \lim\limits ^ \flt A _ {i} $, where $ A _ {i} $ are polyhedral chains, and $ \psi A = \lim\limits ^ \srp A _ {i} $, where $ \psi $ is a linear bijective mapping from the space $ C _ {r} ^ \flt ( E ^ {n)} $ into the space $ C _ {r} ^ \srp ( E ^ {n)} $, and $ \psi C _ {r} ^ \flt $ is dense in $ C _ {r} ^ \srp $ in the sharp norm.

It is not possible to give a correct definition of the boundary $ \partial A $ of a sharp chain [1]; an $ r $- dimensional sharp chain $ X = XA $ is an element of the space $ C ^ {\srp r } ( E ^ {n} ) $ dual to $ C _ {r} ^ \srp ( E ^ {n} ) $; it is a flat cochain, and

$$ | X | \leq | X | ^ \flt \leq | X | ^ \srp , $$

where $ | X | $ is the co-mass of $ X $, while the sharp co-norm $ | X | ^ \srp $ is defined similarly to the flat norm $ | X | ^ \flt $. The co-boundary $ dX $ of a sharp cochain is not necessarily sharp [1], but

$$ | dX | \leq | X | ^ \flt \leq | X | ^ \srp . $$

The Lipschitz constant $ {\mathcal L} ( X) $ of a cochain $ X $ is defined as follows:

$$ {\mathcal L} ( X) = \sup \left \{ \frac{| X \cdot ( T _ {v} A - A ) | }{| A | | v | } \right \} , $$

where the $ A $ are polyhedral chains. For sharp cochains this supremum is finite, and

$$ ( r + 1 ) {\mathcal L} ( X) \leq | X | ^ \srp . $$

Any flat cochain with a finite Lipschitz constant is sharp, and

$$ | X | ^ \srp = \sup \{ | X | ^ \flt ,\ ( r + 1 ) {\mathcal L} ( X) \} , $$

and also

$$ | dX | \leq ( r + 1 ) {\mathcal L} ( X) . $$

Similar concepts are introduced for $ r $- dimensional polyhedral chains in open subsets $ R \subset E ^ {n} $. See also Sharp form.

The sharp norm in the space of additive functions $ \gamma $ whose values are $ r $- vectors is the largest of the semi-norms $ | \cdot | ^ \prime $ which satisfy the conditions:

$ | \gamma | ^ \prime \leq | \gamma | $, where $ | \gamma | $ is the complete variation of $ \gamma $;

$$ | T _ {v} \gamma - \gamma | ^ \prime \leq \ \frac{| v | | \gamma | }{r+} 1 , $$

where $ T _ {v} \gamma ( Q) = \gamma T _ {-} v ( Q) $ is the shift of the function $ \gamma $ by the vector $ v $ of length $ | v | $:

$$ T _ {-} v ( Q) = \{ {q - v } : {q \in Q \subset E ^ {n} } \} ; $$

for each point $ p $ and an arbitrary $ \epsilon $ there exists an $ \eta > 0 $ such that $ | \gamma | ^ \prime \leq \epsilon | \gamma | $ if the support $ \supp t \gamma \subset U _ \eta ( p) $ and $ \gamma ( E ^ {n)} = 0 $.

The sharp norm $ | \gamma | ^ \srp $ is represented as follows:

$$ | \gamma | ^ \srp = \sup _ \omega \int\limits _ {E ^ {n} } \omega d \gamma , $$

where the $ \omega $ are $ r $- dimensional sharp forms for which $ | \omega | ^ \srp \leq 1 $.

References

[1] H. Whitney, "Geometric integration theory" , Princeton Univ. Press (1957)
How to Cite This Entry:
Sharp norm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sharp_norm&oldid=48681
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article