Not my problem . Not worth my thought . Lol . <| sep |> Ummm . If the shape has a volume , let ' s say that means it has a Lesbague measure , in which case , it contains a finite set of open rectangles with a volume as close as we like to the shape ' s volume . Now if we demand that the new shape ' s boundary is close to the old one on the scale of the smallest rectangle edge , then we won ' t have excluded much of the rectangles . Also , I think there is a dual argument for a set of rectangles containing the shape . I don ' t like your L 2 measure of boundary distance because you haven ' t fixed your parameterisation of the boundary , so you might have big changes that only affect a small range of parameter values , and hence only change L 2 slightly .