from the retraction. We describe the details of retraction for an ℓ -simplex σ in the p -dimensional simplicial complex K . This set up is valid for any ℓ , but in particular, we will use the bounds thus obtained for d ≤ ℓ ≤ p when retracting a d -current onto K . We pick a center a ∈ Int( σ ), the interior of σ , and project every x ∈ Int( σ ) \ { a } along the ray ( x − a ) / ‖ x − a ‖ to perimeter( σ ). Denoting this map as φ ( x , a ), we get where φ$_{δ}$ ( x , a ) is a dilation of R ℓ by the factor ∥ $_{∥}$φ ( x , a ) − a ∥ / ‖ x − a ‖ and φ$_{π}$ ( x , a ) is a nonorthogonal projection along ( x − a ) / ‖ x − a ‖ onto τ$_{x}$ , the ( ℓ − 1)-dimensional face of σ containing φ ( x , a ). We denote ˆ r = ∥ $_{∥}$φ ( x , a ) − a ∥ and r = ‖ x − a ‖ . Let E$_{ℓ}$ be the ℓ -hyperplane that contains σ and E$_{ℓ}$$_{−}$$_{1}$ the ( ℓ − 1)-hyperplane that contains τ$_{x}$ . Denote the orthogonal projection of a onto E$_{ℓ}$$_{−}$$_{1}$ by b , and let ˆ h = ‖ b − a ‖ . For any point y = a + ( b − a ) γ with 0 < γ < 1, we get φ ( y , a ) = b . In particular, we consider the point of intersection of line connecting a and b with the ( ℓ − 1)-hyperplane parallel to τ$_{x}$ that contains x . Naming this point y , we define h = ‖ y − a ‖ . Let z ∈ E$_{ℓ}$ denote either normal to τ$_{x}$ at φ ( x , a ) (either of the two possibilities work). Let v$_{2}$ = ( x − a ) / ‖ x − a ‖ , and let v$_{1}$ be the vector in span( z , v$_{2}$ ) that is normal to v$_{2}$ and points into σ . We illustrate this construction on a 3-simplex in Figure 2.3, where the cone of a with face τ is shown in red and the other points and vectors are labeled. We also illustrate the corresponding slice spanned by v$_{1}$ and v$_{2}$ in Figure 2.4. Choose an orthogonal basis { w$_{1}$ , ..., w$_{ℓ}$$_{−}$$_{2}$ } for span( v$_{1}$ , v$_{2}$ ) $^{⊥}$. Note that span( w$_{1}$ , ..., w$_{ℓ}$$_{−}$$_{2}$ ) ⊂ E$_{ℓ}$$_{−}$$_{1}$ . Let w ′ be a unit vector in span( w$_{1}$ , ..., w$_{ℓ}$$_{−}$$_{2}$ ) ⊥ ∩ E$_{ℓ}$$_{−}$$_{1}$ parallel to φ ( x ) − b . 32 φ ( x , a ) = ( φ$_{π}$ ◦ φ$_{δ}$ )( x , a ) , (2.14) Retracting from a center inside a simplex