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

0101200000000000011111111111110000 111101010101010101010101Figure 6: Fattening of a setProperties 6.12• (Ξ, d H ) is path-metric;• d H (A, B) = supx∈lR N |u A (x) − u B (x)| .• each family of compact sets that are contained in a fixed large closed ballin lR N is compact in (Ξ, d H ); so• any closed ball D(A, ρ) := {B | d H (A, B) ≤ ρ} is compact in (Ξ, d H ), andmoreover• by Theorem 6.11, minimal geodesics exist in (Ξ, d H ).6.2.1 An alternative definitionLet D r (x) be the closed ball of center x and radius r > 0 in lR N , and D r = D r (0).We define the fattened set to beA + D r := {x + y | x ∈ A, |y| ≤ r} = ⋃D r (x) = {y | u A (y) ≤ r}.Note that the fattened set is always closed, (since the distance function u A (x) iscontinuous).Example 6.13 In figure 6 we see an example of a set A fattened to r = 1, 2;the set A is the black polygon (and is filled in), whereas the dashed lines in thedrawing are the contours of the fattened sets. 7We can then state the following equivalent definition of the Hausdorff distance:x∈Ad H (A, B) = min{δ > 0 | A ⊂ (B + D δ ), B ⊂ (A + D δ )} .7 The fattened sets are not drawn filled — otherwise they would cover A.44