  We suggest that the curse of dimensionality affecting the similarity-based
search in large datasets is a manifestation of the phenomenon of concentration
of measure on high-dimensional structures. We prove that, under certain
geometric assumptions on the query domain $\Omega$ and the dataset $X$, if
$\Omega$ satisfies the so-called concentration property, then for most query
points $x^\ast$ the ball of radius $(1+\e)d_X(x^\ast)$ centred at $x^\ast$
contains either all points of $X$ or else at least $C_1\exp(-C_2\e^2n)$ of
them. Here $d_X(x^\ast)$ is the distance from $x^\ast$ to the nearest neighbour
in $X$ and $n$ is the dimension of $\Omega$.
