Proof. We must show that there are centers in the set of feasible centers B σ (see the proof sketch of Theorem 3.2.4) which simultaneously achieve the various bounds on the 2( m + n ) relevant currents: T$_{1}$, . . . , T$_{m}$ , ∂T$_{1}$, . . . ∂T$_{m}$ , S$_{1}$, . . . S$_{n}$ , ∂S$_{1}$, . . . , ∂S$_{n}$ . We consider the case of projecting currents from the ℓ -skeleton to the ( ℓ − 1)-skeleton in the ℓ -simplex σ . As in the proof of Theorem 3.2.4, we again use the average bound in Equation (3.9). For each k ∈ Z + and i = 1 , 2 , . . . , m , let Then, using the same average-based argument as in Theorem 3.2.4, we have that H $^{ℓ}$( H$_{T}$$_{i}$$_{,k}$ ) / H $^{ℓ}$( B $_{σ}$) < 1 2 m +2 n (i.e., the size of the set of poorly behaved centers with respect to each T$_{i}$ is a small fraction of the set B σ of possible centers). We similarly define H$_{∂T}$$_{i}$$_{,k}$ , H$_{S}$$_{j}$$_{,k}$ , and H$_{∂S}$$_{j}$$_{,k}$ and obtain the same bound of 1 2 m +2 m on the bad centers. For each k ∈ Z $^{+}$, we are interested in the set of centers which are simultaneously good centers for all currents involved (i.e., points in B σ but not any of the H$_{·}$$_{,k}$ sets). Call this set G$_{k}$ and observe that it has positive measure: Thus for any k > 1 ϵ we have that G$_{k}$ is a nonempty set of possible projection centers which simultaneously attain an expansion bound of at most (2 m + 2 n + ϵ ) ϑ$_{σ}$ for all the pertinent currents. 65 H $^{ℓ}$( G$_{k}$ ) = H ℓ    B σ \$^{}$  m ⋃ i =1 H$_{T}$$_{i}$$_{,k}$ ∪ m ⋃ i =1 H$_{∂T}$$_{i}$$_{,k}$ ∪ n ⋃ i =1 H$_{S}$$_{i}$$_{,k}$ ∪ n ⋃ i =1 H$_{∂S}$$_{i}$$_{,k}$$^{}$     ≥ H $^{ℓ}$( B $_{σ}$) − m ∑ i =1 H $^{ℓ}$( H$_{T}$$_{i}$$_{,k}$ ) − m ∑ i =1 H $^{ℓ}$( H$_{∂T}$$_{i}$$_{,k}$ ) − n ∑ j =1 H $^{ℓ}$( H$_{S}$$_{j}$$_{,k}$ ) − n ∑ j =1 H $^{ℓ}$( H$_{∂S}$$_{j}$$_{,k}$ ) > H $^{ℓ}$( B $_{σ}$) ( 1 − m 2 m + 2 n − m 2 m + 2 n − n 2 m + 2 n − n 2 m + 2 n$^{)}$ = 0 . H$_{T}$$_{i}$$_{,k}$ = { a ∈ B σ ∣ ∣ ∣ ∫ σ J$_{d}$φ ( x , a ) d ‖ T$_{i}$ ‖ ( x ) > ( 2 m + 2 n + 1 k$^{)}$ ϑ$_{σ}$ M( T$_{i}$ )$^{}}$ .