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

CutΩ(The arrows represent the distance R(s) from y(s) ∈ ∂Ω to the cutlocus).Figure 9: Smooth contour and cutlocus.6.3.7 Pullback of the metric on smooth contoursIf Ω is smooth but not convex, then the above formula holds up to the cutlocus.Definition 6.22 The cutlocus ( a.k.a. the external skeleton) is the set ofpoints x ∉ Ω such that there are two (or more) different points y 1 , y 2 ∈ Ω ofminimum distance from x to Ω, that is,|x − y 1 | = |x − y 2 | = u Ω (x) .We define a function R(s) : [0, L] → [0, ∞] that measures the distance from ∂Ωto the cutlocus Cut; in turn, we can parameterize Cut asCut = {ψ(R(s), s) | s ∈ [0, L], R(s) < ∞} ;R(s) is locally Lipschitz where it is finite (by results in Itoh and Tanaka [25], Liand Nirenberg [32]). The “polar” change of coordinates ψ (defined in (6.3)) is adiffeomorphism between the sets{(ρ, s) s ∈ [0, L], 0 < ρ < R(s)} ↔ lR 2 \ (Ω ∪ Cut) .See fig. 9.The pullback metric for deformations of the contour ∂Ω is∫ [ ∫ ]R(s)〈h, k〉 =(ϕ ′ (ρ)) 2 (1 + ρκ(s)) dρ α(s)β(s) ds∂Ω06.3.8 ConclusionIn this case we have found (a posteriori) a Riemannian metric of closed embeddedplanar curves, and we know the structure of the completion, and the completionadmits minimal geodesics. On the down side, the completion is not really “asmooth Riemannian manifold”. For example, it is difficult to study the minimalgeodesics, and to prove any property about them. Anyway, the fact that N c islocally compact and the regularity property 6.18 of L p spaces suggest that itmay be possible to “shoot geodesics” (in some weak form). Unfortunately thetopology has too few open sets to be used for shape optimization: many simpleenergy functionals are not continuous.50