Cut locus of a separating fractal set in a Riemannian manifold

We study the geometry of the cut locus of a separating fractal set A in a R iemannian manifold. In particular, we prove that every point of A is a limi t point of the cut locus C(A) of A, and the Hausdorff dimension of C(A) is greater than or equal to that of A. Furthermore, we study the cut locus of the well-known Koch snowflake, and show the Hausdorff dimension of its cut locus is log 6/log 3 which is greater than the Hausdorff dimension, log 4/l og 3, of the Koch snowflake itself. We also give another example for which the Hausdorff dimension of the cut locus stays the same. These two new exam ples are new fractal objects which are of interest on their own right.