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 $Ω$ and the dataset $X$, if $Ω$ 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 $Ω$.