Does there exist a uniform autohomeomorphism $\mathbb{R} ^ 2$ transforming the disc $\{(x, y) \mid x ^ 2 + y ^ 2 \leq 1\}$ into Koch snowflake? The Koch snowflake is a closed subset of the plane bounded by the Koch curve.
I understand that this subspace itself is uniformly homeomorphic to a disc.
If instead of the Koch snowflake we take a triangle, a rectangle and, in general, any convex figure containing a unit disk, then the desired autohomeomorphism of the plane is constructed using some stretching of the rays going out from the center of the disk.
Of course, according to Schoenflies's theorem, there is an autohomeomorphism of the large disk that turns the small disk in its center into a coch snowflake. Since the disk is compact, the same homeomorphism is uniform.