Cut locus of a separating fractal set in a Riemannian manifold (Q1283229)

The cut locus \(C(A)\) of a set \(A\) in a Riemannian manifold can be defined to be the set of points \(x\) such that there is a geodesic \(\gamma\) from a point \(p\in A\) to \(x\) where any point \(z\in\gamma\) between \(p\) and \(x\) minimises the distance between \(A\) and \(z\), but \(\gamma\) stops doing so beyond \(x\). For a point set \(A\), the corresponding cut locus becomes a well-known Voronoi diagram. The authors study the geometry of the cut locus of a separating fractal set \(A\) in a Riemannian manifold. They prove that every point of \(A\) is a limit of the cut locus \(C(A)\) and the Hausdorff dimension of \(C(A)\) is greater than or equal to the dimension of \(A\). For instance, the Koch snowflake has the cut locus of dimension \(\log 6/\log 3\), greater than \(\log 4/\log 3\), the dimension of the Koch snowflake itself. Another non-trivial example of \(A\) where the dimension of \(A\) is equal to that of \(C(A)\) is given. These two examples define new fractal objects that may be of interest on their own right.