Embeddings and non-approximability of geometric problems

In this paper we present two results. Our first theorem shows that "sparse" (see definition later) metric spaces, with distances 1 and 2, are isometrically embeddable into lowdimensional loo norm. A similar in "spirit" result was earlier proved for lp norms with finite p by Trevisan [Tre97]. Our result is obtained using a different technique (Bourrgain's sampling approach) and could not have been achieved using the techniques of [Tre97]. In the second part of the paper we apply the aforementioned embeddings to show non-approximability results for several geometric problems in Ip norms with logarithmic dimension, namely TSP, k-median and rain-sum k-clustering. In particular, we provide the first known constant factor hardness for the rain-sum k-clustering problem.