The Hilbert-curve satisfies the Hölder-condition, thus defining an upper limit for the euclidian distance of two points in the plane in dependency of their distance on the Hilbert curve.
What I'm looking for is the inverse statement: Is there an upper limit for the distance of two points on the Hilbert curve given their distance on the plane?
After experimenting a little with this widget I assume that there is no such limit, but I wanted to ask somebody who really knows about this mathematical discipline.
