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 l1.

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 A0 and A1 generated by self-similar iterated function systems F0 and F1. More generally, we construct a family of fractal sets (Aw) parameterized by the Cantor set C = {0,1}; for a given point w in C the set Aw is the invariant set for the graph directed IFS obtained by contracting at the mth level with either F0 or F1 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 Aw at abscissa x. By construction, the x-slices of the family of sets (Aw) 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 RN", Bull. London Math. Soc. 34 (2002), 667-676.

[2] J. R. Lee, M. Mendel and A. Naor, "Metric structures in l1: 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.