content_image/307932.jpg content_image/307940.jpg
Figure 3.3). We wish to show 

<img src='content_image/307932.jpg'>

<img src='content_image/307940.jpg'>

where P$_{δ}$ is a simplicial approximation to T on some complex K$_{δ}$ with F ( P$_{δ}$ − T ) < δ . 

This goal prevents us from simply using the simplicial deformation theorem to obtain P$_{δ}$ since we may end up with the situation in Figure 3.3. Instead, we use a polyhedral approximation to T which guarantees that the mass increases by at most δ (i.e., M( P$_{δ}$ ) < M( T ) + δ rather than the simplicial deformation theorem bound M( P$_{δ}$ ) < C$_{1}$ M( T ) + C$_{2}$ M( ∂T ) with constants bounded away from 1). 

The next step is to take an optimal (possibly nonintegral) decomposition of T and approximate it with polyhedral chains (see Figure 3.4). That is, approximate the de- composition T = X + ∂S with polyhedral X$_{δ}$ and S$_{δ}$ . If these approximations naturally form a decomposition (not necessarily optimal) of P$_{δ}$ (i.e., P$_{δ}$ = X$_{δ}$ + ∂S$_{δ}$ ), then we would have F$_{K}$$_{δ}$ ( P$_{δ}$ ) ≤ M( X$_{δ}$ ) + M( S$_{δ}$ ) < F ( T ) + 2 δ for any complex K$_{δ}$ containing P$_{δ}$ , X$_{δ}$ , and S$_{δ}$ . This of course implies Equation (3.1). 

However (as in Figure 3.5), we need not have P$_{δ}$ = X$_{δ}$ + ∂S$_{δ}$ . Since we obtained these quantities by polyhedral approximation, it turns out that the extent to which this 

