On the nearest neighbor of the nearest neighbor in multidimensional continuous and quantized space
The probability that an entity in a set of entities uniformly distributed in space is the nearest neighbor of its nearest neighbor is evaluated for generic distances in a multidimensional environment. Such an expression is then specialized for systems with norm-based distances and for systems with quantized norm-based distance. Examples for scalar products and sup-norm are derived. When appticable, invariances with respect to the underlying distance and entities density are highlighted. Dimensionality effects are investigated.
nearest neighbor dimensionality effect euclidean distances poisson point processes quantized distances