# Sharp norm

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 $| A | ^ \sharp$ is expressed as follows:

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

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

One has

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

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

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

$$| A | ^ \sharp \leq | A | ^ \flat ;$$

if $r = 0$, then $| A | ^ \sharp = | A | ^ \flat$.

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

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

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 $A = \lim\limits ^ \flat A _ {i}$, where $A _ {i}$ are polyhedral chains, and $\psi A = \lim\limits ^ \sharp A _ {i}$, where $\psi$ is a linear bijective mapping from the space $C _ {r} ^ \flat ( E ^ {n)}$ into the space $C _ {r} ^ \sharp ( E ^ {n)}$, and $\psi C _ {r} ^ \flat$ is dense in $C _ {r} ^ \sharp$ 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 $C ^ {\sharp r } ( E ^ {n} )$ dual to $C _ {r} ^ \sharp ( E ^ {n} )$; it is a flat cochain, and

$$| X | \leq | X | ^ \flat \leq | X | ^ \sharp ,$$

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

$$| dX | \leq | X | ^ \flat \leq | X | ^ \sharp .$$

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

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

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

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

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 $| \gamma | ^ \sharp$ is represented as follows:

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

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

#### 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=52359
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article