Proof. Following Equation (2.14), we seek bounds on Dφ$_{δ}$ ( x , a ) and Dφ$_{π}$ ( x , a ). Since Dφ$_{δ}$ ( x , a ) simply scales by ˆ r/r = ˆ h/h , the expansion of d -volume of any d -hyperplane by Dφ$_{δ}$ ( x , a ) is by a factor of ( ˆ h/h ) $^{d}$. On the other hand, bounding the dilation that Dφ$_{π}$ ( x , a ) can cause in d -hyperplanes is a little more involved. We seek a bound on for all ℓ × d matrices U . Using the generalized Pythagorean theorem [29, Section 1.5], we get where submatrix U$_{λ}$ consists of the d rows of U specified by the set of index maps Λ given as A similar result holds for det(( Dφ$_{π}$ ( x , a ) U ) $^{T}$( Dφ$_{π}$ ( x , a ) U )), with the functions f con- sidered mapping [1 , ..., d ] to [1 , ..., ℓ − 1]. Observe that multiplying by Dφ$_{π}$ ( x , a ) (Equation 2.16) just scales the first row of U by α and removes the last row. Thus α det( U$_{λ}$ ) ≥ det(( Dφ$_{π}$ ( x , a ) U )$_{λ}$), which implies that α is a bound on the ratio in Equation (2.17). Thus we have that holds for all x and a in σ , where diameter( σ ) is the diameter of the ℓ -simplex σ . Next we describe a bound on the integral of J$_{d}$φ ( x , a ) over the entire ℓ -simplex, for a fixed center a . We will find that this bound is independent of the position of a . Recall that perimeter( σ ) and P( σ ) denote the perimeter of ℓ -simplex σ and the 35 J$_{d}$φ ( x , a ) = $^{(}$ˆ r r$^{)}$$^{d}$$^{∥}$ $_{∥}$φ ( x , a ) − a ∥ ‖ b − a ‖ = $^{(}$ˆ h h$^{)}$$^{d}$$^{∥}$ $_{∥}$φ ( x , a ) − a ∥ ‖ b − a ‖ ≤ $^{(}$ˆ h h$^{)}$$^{d}$ diameter( σ ) ˆ h λ ∈ Λ ≡ { f | f : [1 , ..., d ] → [1 , ..., ℓ ] , f is one to one and increasing } . det( U $^{T}$U ) = ∑ λ ∈ Λ (det( U$_{λ}$ )) 2 $^{√}$det( ( Dφ$_{π}$ ( x , a ) U ) $^{T}$( Dφ$_{π}$ ( x , a ) U ) ) $^{√}$det( U $^{T}$U ) (2.17)