LOG-LIPSCHITZ EMBEDDINGS OF HOMOGENEOUS SETS WITH SHARP LOGARITHMIC EXPONENTS AND SLICING PRODUCTS OF BALLS

作者：Robinson James C^*

来源：PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 2014, 142(4): 1275-1288.

DOI：10.1090/s0002-9939-2014-11852-1

If X is a compact subset of a Banach space with X - X homogeneous (equivalently 'doubling' or with finite Assouad dimension), then X can be embedded into some R-n (with n sufficiently large) using a linear map L whose inverse is Lipschitz to within logarithmic corrections. More precisely, there exist c, alpha > 0 such that
c parallel to x-y parallel to/vertical bar log parallel to x-y parallel to vertical bar(alpha) <= vertical bar Lx - Ly vertical bar <= c parallel to x-y parallel to for all x,y epsilon X, parallel to x-y parallel to < delta,
for some delta sufficiently small. It is known that one must have alpha > 1 in the case of a general Banach space and alpha > 1/ 2 in the case of a Hilbert space. It is shown in this paper that these exponents can be achieved.
While the argument in a general Banach space is relatively straightforward, the Hilbert space case relies on the fact that the maximum volume of a hyperplane slice of a k-fold product of unit volume N-balls is bounded independent of k (this provides a 'qualitative' generalisation of a result on slices of the unit cube due to Hensley (Proc. AMS 73 (1979), 95-100) and Ball (Proc. AMS 97 (1986), 465-473)).

• 出版日期2014-4