High dimensionality can pose severe difficulties, widely recognized as different aspects of the curse of dimensionality. In this paper we study a new aspect of the curse pertaining to the distribution of k-occurrences, i.e., the number of times a point appears among the k nearest neighbors of other points in a data set. We show that, as dimensionality increases, this distribution becomes considerably skewed and hub points emerge (points with very high k-occurrences). We examine the origin of this phenomenon, showing that it is an inherent property of highdimensional vector space, and explore its influence on applications based on measuring distances in vector spaces, notably classification, clustering, and information retrieval.