Sharp norm

in the space of -dimensional polyhedral chains

The largest semi-norm which, for any cell of volume , satisfies the inequalities

where is the cell obtained by shifting by a vector of length .

If , the sharp norm is expressed as follows:

where is the flat norm of the chain , and the infimum is taken over all shifts .

One has

if , then .

The completion of the space is the separable Banach space , whose elements are known as -dimensional sharp chains. For any -dimensional polyhedral chain and any vector ,

where is the chain obtained by shifting by the vector of length . 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 , where are polyhedral chains, and , where is a linear bijective mapping from the space into the space , and is dense in in the sharp norm.

It is not possible to give a correct definition of the boundary of a sharp chain [1]; an -dimensional sharp chain is an element of the space dual to ; it is a flat cochain, and

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

The Lipschitz constant of a cochain is defined as follows:

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

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

and also

Similar concepts are introduced for -dimensional polyhedral chains in open subsets . See also Sharp form.

The sharp norm in the space of additive functions whose values are -vectors is the largest of the semi-norms which satisfy the conditions:

, where is the complete variation of ;

where is the shift of the function by the vector of length :

for each point and an arbitrary there exists an such that if the support and .

The sharp norm is represented as follows:

where the are -dimensional sharp forms for which .

References

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