Recent research has shown that machine learning systems, including state-of-the-art deep neural networks, are vulnerable to adversarial attacks. By adding to the input object an imperceptible amount of adversarial noise, it is highly likely that the classifier can be tricked into assigning the modified object to any desired class. It has also been observed that these adversarial samples generalize well across models. A complete understanding of the nature of adversarial samples has not yet emerged. Towards this goal, we present a novel theoretical result formally linking the adversarial vulnerability of learning to the intrinsic dimensionality of the data. In particular, our investigation establishes that as the local intrinsic dimensionality (LID) increases, 1-NN classifiers become increasingly prone to being subverted. We show that in expectation, a k-nearest neighbor of a test point can be transformed into its 1-nearest neighbor by adding an amount of noise that diminishes as the LID increases. We also provide an experimental validation of the impact of LID on adversarial perturbation for both synthetic and real data, and discuss the implications of our result for general classifiers.