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_infinity 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}^infinity; 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) is equivariant with respect to the quotient relation defining L. Thus the embedding is well-defined. The 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.