** THE COMPACTA HYPERSPACE OF THE UNIT INTERVAL AND BI-LIPSCHITZ
EMBEDDINGS **

The compacta hyperspace K(X) of a complete metric space X is the set of nonempty compact subsets of X equipped with the Hausdorff metric. In [4] we proved the following theorem:

(*) K([0,1]) admits no bi-Lipschitz embedding into any uniformly convex Banach space.

This contrasts with a celebrated result of Schori and West [3] stating that K([0,1]) is homeomorphic with the Hilbert cube.

We sketch the proof of (*).

The Laakso space L is a certain self-similar doubling series-parallel
path metric space which admits no bi-Lipschitz embedding into any
uniformly convex Banach space. Laakso [1] studied the space L in
connection with A_{∞} deformations of Euclidean
geometry. Lee, Mendel and Naor [2] used L as an example in algorithmic
network theory and nonlinear geometric functional analysis related to
the failure of the Johnson-Lindenstrauss dimension reduction theorem
in l_{1}.

We prove (*) as a corollary of

(**) There is a bi-Lipschitz embedding of L into K([0,1]).

The following figure shows a pair of planar fractals A_{0} and A_{1}
generated by self-similar iterated function systems F_{0} and F_{1}. More
generally, we construct a family of fractal sets (A_{w}) parameterized
by the Cantor set C = {0,1}^{∞}; for a given point w in C the
set A_{w} is the invariant set for the graph directed IFS obtained by
contracting at the mth level with either F_{0} or F_{1} according to the
mth coordinate of w. The figures correspond to the case w=(0,0,0,...)
and w=(1,1,1,...) respectively.

The embedding of L in K([0,1]) is defined as follows. Represent L as a
quotient of C x [0,1]. The image of a point (w,x) is defined as the
vertical slice of A_{w} at abscissa x. By construction, the x-slices of
the family of sets (A_{w}) are invariant with respect to the quotient
relation defining L. Thus the embedding is well-defined.
Self-similarity of the various defining constructions yields that the
embedding is bi-Lipschitz. Details can be found in [4].

[1] T. J. Laakso, "Plane with A_infinity weighted metric not
bi-Lipschitz embeddable to R^{N}", *Bull. London Math.
Soc.* ** 34 ** (2002), 667-676.

[2] J. R. Lee, M. Mendel and A. Naor, "Metric structures in l_{1}:
dimension, snowflakes and average distortion", * European J.
Combin. * ** 26 ** (2005) no. 8, 1180-1190.

[3] R. M. Schori and J. E. West, "The hyperspace of the closed unit
interval is a Hilbert cube", *Trans. Amer. Math. Soc.* **
213 ** (1975), 217-235.

[4] J. T. Tyson, "Bi-Lipschitz embeddings of hyperspaces of compact
sets", *Fund. Math.* ** 187 ** (2005), no. 3, 229-254.