We show that the Hausdorff metric over constant-size pointsets in constant-dimensional Euclidean space admits an embedding into constant-dimensional `∞ space with constant distortion. More specifically for any s, d ≥ 1, we obtain an embedding of the Hausdorff metric over pointsets of size s in d-dimensional Euclidean space, into `s ∞ with distortion sO(s+d). We remark that any metric space M ad...

We investigate when the Hutchinson operator associated with an iterated function system is continuous. The continuity with respect to both the Hausdorff metric and Vietoris topology is carefully considered. An example showing that the Hutchinson operator on the hyperspace of nonempty closed bounded sets need not be Hausdorff continuous is given. Infinite systems are also discussed. The work cla...

We characterize the coordinate functions of Hilbert's spacelling curve using a directed-graph iterated function system and use this to analyze their fractal properties. In particular, we show that both coordinate functions have graphs of Hausdor dimension 3 2 and level sets of dimension 1 2 .

A Hausdorff topological group G is minimal if every continuous isomorphism f : G → H between G and a Hausdorff topological group H is open. Clearly, every compact Hausdorff group is minimal. It is well known that every infinite compact Hausdorff group contains a non-trivial convergent sequence. We extend this result to minimal abelian groups by proving that every infinite minimal abelian group ...

The concern of this paper is with effective packing dimension. This concept can be traced back to the work of Borel and Lebesgue who studied measure as a way of specifying the size of sets. Carathéodory later generalized Lebesgue measure to the n-dimensional Euclidean space, and this was taken further by Hausdorff [Hau19] who generalized the notion of s-dimensional measure to include non-intege...

Given a curve f and a surface S, how hard is it to find a simple curve f ′ ⊆ S that is the most similar to f? We introduce and study this simple curve embedding problem for piecewise linear curves and surfaces in R and R, under Hausdorff distance, weak Fréchet distance, and Fréchet distance as similarity measures for curves. Surprisingly, while several variants of the problem turn out to have p...

Consider two (n−1)-dimensional manifolds, S and S in R. We say that they are normalcompatible when the closest projection of each one onto the other is a homeomorphism. We give a tight condition under which S and S are normal-compatible. It involves the minimum feature size of S and of S and the Hausdorff distance between them. Furthermore, when S and S are normal-compatible, their Frechet dist...

I. J. Good [Proc. Cambridge Philos. Soc. 37 (1941), pp. 199–228] showed that the set of irrational numbers in ( 0 , 1 stretchy="...

We show that Hausdorff measures of different dimensions are not Borel isomorphic; that is, the measure spaces (R, B, H) and (R, B, H) are not isomorphic if s 6= t, s, t ∈ [0, 1], where B is the σ-algebra of Borel subsets of R and H is the d-dimensional Hausdorff measure. This answers a question of B. Weiss and D. Preiss. To prove our result, we apply a random construction and show that for ever...