Metrics of curves in shape optimization and analysis - Andrea Carlo ...

and similarly P is the evaluationof the polynomialP = p ( ‖h 1 ‖ 0 , ‖h 2 ‖ 0 , ‖c ′ 1‖ 0 , ‖c ′ 2‖ 0 , 1/(len(c 1 ) len(c 2 )) )p(x 1 , x 2 , x 3 , x 4 , x 5 ) = (π +1/2)(x 1 +x 2 )+2π 3 (x 1 x 3 +x 2 x 4 )(x 3 +x 4 )x 5 (10.31)(that has constant positive coefficients).Proof. Fix c i immersed, and h i ∈ T ci M for i = 1, 2; let L i = len(c i ); letk i := P ci Π ci h i (s)for simplicity. We rewrite this in the convolutional form∫k i (s) := K i (s − ŝ)h i (ŝ) dŝ (10.32)c iwhen integrals are performed in arc parameter, and the kernel (following (10.21))isK i (s) := − s + 1 for s ∈ [0, L i ]L i 2and K i is extended periodically. By substituting and integrating on only oneperiod of K i ,∫ s( 1k i (s) =s−L i2 − s − ŝ )h i (ŝ) dŝ .L iWe can then prove easily (10.29): indeed by the convolutional representation(10.32)∫ len(c)|(P c Π c h)(t)| ≤ ‖h‖ 0 ∣ − slen(c) + 1 2∣ ds ≤ len(c)‖h‖ 0and instead, deriving and applying (iv) from lemma 10.25,0|(P c Π c h) ′ (θ)| = |Π c h(θ)| |c ′ (θ)| ≤ 2‖h‖ 0 ‖c ′ ‖ 0 .To prove (10.30), we write (10.32) in θ parameter, as was done in theDefinition 10.7; we need the run-length functions l i : lR → lRl i (τ) :=∫ τ0|c ′ i(x)| dxso that, setting s = l i (τ) and ŝ = l i (θ) we can writek i (τ) =∫ ττ−2π( 12 − l i(τ) − l i (θ))h i (θ)|c ′Li(θ)| dθi75