Mass and co-mass

Adjoint norms (cf. Norm) in certain vector spaces dual to each other.

1) The mass of an $ r $- vector $ \alpha $, i.e. an element of the $ r $- fold exterior product of a vector space, is the number

$$ | \alpha | _ {0} = \ \inf \ \left \{ {\sum _ { i } | \alpha _ {i} | } : {\alpha = \sum {\alpha _ {i} } ,\ \alpha _ {i} \ \textrm{ simple } r \textrm{ - vectors } } \right \} . $$

The co-mass of an $ r $- covector $ \omega $ is the number

$$ | \omega | _ {0} = \ \sup _ \alpha \{ {| \omega \cdot \alpha | } : { \alpha \textrm{ a simple } r \textrm{ - vector } , | \alpha | = 1 } \} . $$

Here $ | \cdot | $ is the standard norm of an $ r $- vector and $ \omega \cdot \alpha $ is the scalar product of a vector and a covector.

The mass $ | \alpha | _ {0} $ and the co-mass $ | \omega | _ {0} $ are adjoint norms in the spaces of $ r $- vectors $ V _ {[} r] $ and $ r $- covectors $ V ^ {[} r] $, respectively. In this connection:

a) $ | \omega | _ {0} = \sup _ \alpha \{ {| \omega \cdot \alpha | } : {| \alpha | _ {0} = 1 } \} $, $ | \alpha | _ {0} = \sup _ \alpha \{ {| \omega \cdot \alpha | } : {| \omega | _ {0} = 1 } \} $;

b) $ | \alpha | _ {0} \geq | \alpha | $, $ | \omega | _ {0} \geq | \omega | $, and equalities hold if and only if $ \alpha $( $ \omega $) is a simple $ r $-( co)vector;

c) $ | \alpha \lor \beta | _ {0} \leq | \alpha | _ {0} | \beta | _ {0} $, $ | \omega \lor \zeta | _ {0} \leq B | \omega | _ {0} | \zeta | _ {0} $ for exterior products $ \lor $, where for a simple multi-covector $ \omega $( or $ \zeta $) $ B = 1 $, and, in general, $ B = ( _ {\ r } ^ {r+} s ) $ if $ \omega \in V ^ {[} r] $ and $ \zeta \in V ^ {[} s] $;

d) $ | \omega \wedge \alpha | _ {0} \leq \widetilde{B} | \omega | _ {0} | \alpha | _ {0} $ for inner products $ \wedge $, where $ \widetilde{B} = 1 $ for $ r \geq s $ and $ \widetilde{B} = ( _ {r} ^ {s} ) $ for $ r \leq s $, $ \omega \in V ^ {[} r] $ and $ \alpha \in V _ {[} s] $.

These definitions enable one to define the mass and co-mass for sections of fibre bundles whose standard fibres are $ V ^ {[} r] $ and $ V _ {[} r] $. For example, the co-mass of a form $ \omega $ on a domain $ G \subset E ^ {n} $ is

$$ | \omega | _ {0} = \ \sup \{ {| \omega ( p) | _ {0} } : {p \in G } \} . $$

2) The mass of a polyhedral chain $ A = \sum {a _ {i} } \sigma _ {i} ^ {r} $ is

$$ | A | = \sum | a _ {i} | | \sigma _ {i} ^ {r} | , $$

where $ | \sigma _ {i} ^ {r} | $ is the volume of the cell $ \sigma _ {i} ^ {r} $. For arbitrary chains the mass (finite or infinite) can be defined in various ways; for flat chains (see Flat norm) and sharp chains (see Sharp norm) these give the same value to the mass.

3) The co-mass of a (flat, in particular, sharp) cochain $ X $ is defined in the standard way:

$$ | X | = \ \sup _ {A \neq 0 } \ \frac{| X \cdot A | }{| A | } , $$

where $ A $ is a polyhedral chain and $ X \cdot A $ is the value of the cochain $ X $ on the chain $ A $.

For references see Flat norm.

Comments

A simple $ r $- vector $ \alpha $ is an element of the form $ \alpha = \beta _ {1} \lor \dots \lor \beta _ {r} $ in the $ r $- fold exterior product $ V _ {[} r] $ of a vector space $ V $. Here "" denotes exterior product and $ \beta _ {1} \dots \beta _ {r} \in V $.

References

[a1]	H. Federer, "Geometric measure theory" , Springer (1969) pp. Sect. 1.8 MR0257325 Zbl 0176.00801