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

6.1.1 Length induced by a distanceIn this subsection (M, d) is a generic metric space.Definition 6.1 (Length by total variation) Define the length of a continuouscurve γ : [α, β] → M, by using the total variationn∑Len d γ := sup d ( γ(t i−1 ), γ(t i ) ) t 1t 6t 4.Ti=1t 0 t 2 t5where the supremum is carried out over all finite subsets T = {t 0 , · · · , t n } ⊂[α, β] and t 0 ≤ · · · ≤ t n .Definition 6.2 (The induced geodesic distance)d g (x, y) := infγ Lend γ (6.1)the infimum is in the class of all continuous γ in M connecting x to y.Note that d g (x, y) < ∞ iff x, y may be connected by a Lipschitz path.6.1.2 Minimal geodesicsDefinition 6.3 If in the definition (6.1) there is a curve γ ∗ providing theminimum, then γ ∗ is called a minimal geodesic connecting x to y.Proposition 6.4 In Riemann and Finsler manifolds, the integral length Len(γ)that we defined in Section 3.6.1 coincides with the total variation length Len d γthat we defined in Definition 6.1. As a consequence, d = d g : we say that d ispath-metric.In general, though, it is easy to devise examples where d ≠ d g .Example 6.5 The set M = S 1 ⊂ lR 2 is a metric space, ifassociated with d(x, y) = |x − y| (that is represented as a dottedsegment in the picture); in this case, d g (x, y) = | arg(x)−arg(y)|(that is represented as a dashed arc in the picture).6.1.3 Existence of geodesics and of averageProposition 6.6 If for a ρ > 0 the closed ballD g (x, ρ) := {y | d g (x, y) ≤ ρ}is compact, then x and any y ∈ D g may be connected by a geodesic.Proposition 6.7 If for a x ∈ M and all ρ > 0, D g (x, ρ) is compact, then thedistance-based average (that was defined in 2.4) exists.The proofs may be found in [15].t 342