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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 high-dimensional vector space, and explore its influence on applications based on measuring distances in vector spaces, notably classification, clustering, and information retrieval.