Flat norm

of an $ r $- dimensional polyhedral chain $ A $ in the space $ E ^ {n} $

The norm $ | A | ^ \flat $ which is defined as follows:

$$ | A | ^ \flat = \ \inf \{ | A - \partial D | + | D | \} , $$

where $ | C | $ is the mass of the chain $ C $( cf. Mass and co-mass), $ \partial C $ is its boundary, and the infimum is taken over all $ ( r+ 1) $- dimensional polyhedral chains. The properties of the flat norm are

$$ | a A | ^ \flat = | a | | A | ^ \flat ,\ \ | A + B | ^ \flat \leq | A | ^ \flat + | B | ^ \flat , $$

$$ | A | ^ \flat = 0 \iff A = 0,\ | A | ^ \flat \leq | A |,\ | \sigma | ^ \flat = | \sigma | $$

for any cell $ \sigma $; if $ \pi $ is the projection of $ E ^ {n} $ on some plane, one has $ | \pi A | ^ \flat \leq | A | ^ \flat $.

The completion of the linear space of polyhedral chains $ C _ {r} ( E ^ {n} ) $ is a separable Banach space, $ C _ {r} ^ \flat ( E ^ {n} ) $; its elements are known as $ r $- dimensional flat chains, to each of which an infinite or a finite mass can be assigned: $ | A | _ \flat = \inf \{ {\lim\limits \inf } _ {i \rightarrow \infty } | A _ {i} |, A _ {i} \rightarrow _ \flat A \textrm{ as polyhedral chains } \} $.

The boundary $ \partial $ of a flat chain is also defined by passing to the limit; it is a continuous operation, and

$$ | \partial A | ^ \flat \leq | A | ^ \flat ,\ \ | A | ^ \flat = \inf \{ | A - \partial D | ^ \flat + | D | ^ \flat \} . $$

The flat norm is the largest of the semi-norms $ | \cdot | ^ \prime $ on $ C _ {r} ( E ^ {n} ) $ in which all cells $ \sigma $ satisfy the inequalities: $ | \sigma ^ {r} | ^ \prime \leq | \sigma ^ {r} | $, $ | \partial \sigma ^ {r+ 1 } | ^ \prime \leq | \sigma ^ {r+ 1 } | $. An $ r $- dimensional flat cochain $ X $ is a linear function of $ r $- dimensional flat chains $ A $( denoted by $ X \cdot A $) such that, for a given $ N $,

$$ | X \cdot A | \leq N | A | ^ \flat , $$

where $ | X | $ is the co-mass of $ X $. It is an element of the non-separable space $ C ^ {\flat r } ( E ^ {n} ) $ dual to $ C _ {r} ^ \flat ( E ^ {n} ) $. The flat norm $ | X | ^ \flat $ of a flat cochain $ X $ is defined in the standard manner:

$$ | X | ^ \flat = \ \sup _ {| A| ^ \flat = 1 } | X \cdot A |, $$

so that

$$ | A | ^ \flat = \ \sup _ {| X| ^ \flat = 1 } | X \cdot A |,\ \ | X \cdot A | \leq | X | ^ \flat | A | ^ \flat , $$

and

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

For the co-boundary $ dX $ of a flat chain (defined by the condition $ dX \cdot A = X \cdot dA $) one has:

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

so that

$$ | X | ^ \flat = \ \sup \{ | X |, | dX | \} . $$

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

References

[1]	H. Whitney, "Geometric integration theory" , Princeton Univ. Press (1957) MR0087148 Zbl 0083.28204