a P$_{δ}$ X$_{δ}$ S$_{δ}$ Figure 3.5: A possible polyhedral approximation of the decomposition shown in Fig- ure 3.1. Note that P$_{δ}$ = X$_{δ}$ + ∂S$_{δ}$ . Figure 3.3). We wish to show 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}$$_{δ}$ ( P$_{δ}$ ) ≤ M( X$_{δ}$ ) + M( S$_{δ}$ ) < F ( T ) + 2 δ for any complex K$_{δ}$ containing P$_{δ}$ , X$_{δ}$ , and S$_{δ}$ . This of course implies Equation (3.1). However (as in Figure 3.5), we need not have P$_{δ}$ = X$_{δ}$ + ∂S$_{δ}$ . Since we obtained these quantities by polyhedral approximation, it turns out that the extent to which this 56 lim δ ↓ 0 F$_{K}$$_{δ}$ ( P$_{δ}$ ) = F ( T ) (3.1)