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On embeddings of finite metric spaces

Let 〈X, q〉 be a finite metric space, and for a natural number d, let ℝ<sup>d</sup> be the real d-dimensional vector space endowed with its usual Euclidean metric. We interested in estimations for d such that 〈X, q〉 can be "embedded" in some sense into ℝ<sup>d</sup>. This classical topic of functional analysis recently has received renewed impetus motivated by several problems of theoretical computer science. We will recall some of these problems which also help us to find the "good" notion of embeddings and announce some recently obtained related results.