text: A major open problem in the field of metric embedding is the existence of dimension reduction for @math -point subsets of Euclidean space, such that both distortion and dimension depend only on the doubling constant of the pointset, and not on its cardinality. In this paper, we negate this possibility for @math spaces with @math . In particular, we introduce an @math -point subset of @math with doubling constant O(1), and demonstrate that any embedding of the set into @math with distortion @math must have @math .
  @cite_12: A Lipschitz embedding of a mctric space (X, d) into another one (Y, 8) is an application : X -»• Y such that : 3 1. B 6 ]0, + oo [. Vx. x ' e X, Ad(x, x ' ) 6( (x), f x' ) Bd(x, x'). We dcscribe here three mcthods to obtain Lipschitz embeddings of the metric space (R*, || ||) into some metric space (R", || || ). The third method allows us to minimize, for k = 1, the rank ofsuch an embedding (i.c. to obtain the minimal value of the integer n).
summary: There are several results on embedding metric spaces with low intrinsic dimension into low dimensional normed space with low distortion: Assouad @cite_12 showed that the snowflakes of doubling metrics For @math , an @math -snowflake of a metric @math is the metric @math , that is, all distances are taken to power @math . embed with constant distortion into constant dimensional Euclidean space. In particular, this is a positive answer to question:d for this special case.