DISTANCES IN BIOLOGY Michel Deza Ecole Normale Superieure, Paris The abstraction of measurement in terms of the mathematical notions distance, similarity, metric, etc. goes back to M.Fr´echet (1906) and F.Hausdorff (1914). Given a set X, a distance (or dissimilarity) on it is a real-valued function d defined on X × X such that d(x, y) ≥ 0, d(x, x) = 0, and d(x, y)=d(y, x) (symmetry) holds for all x, y in X. A similarity is a symmetric function defined on X × X such that s(x, y) ≥ 0 and s(x, y) ≤ s(x, x) holds for all x, y in X, with equality if and only if x coincides with y. A metric is a distance such that d(x, y) = 0 holds only in case x = y, and the triangle inequality d(x, y) ≤ d(x, z) + d(z, y) holds for all x, y, z ∈ X. We present a selection of important notions related to distances/similarities and their uses in Biology.

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