THE AVERAGE DISTANCE BETWEEN TWO POINTS

On Approximating the Average Distance Between

On the Distribution of the Distance Between Two Multivariate Normally Distributed Points

Motivated by the problem of cluster identification, we consider two multivariate normally distributed points. We seek to find the distribution of the squared Euclidean distance between these two point. Consequently, we find the corresponding distribution in the general case. We then reduce this distribution for special cases based on the mean and covariance.

A Distance Between Channels: the average error of mismatched channels

Two channels are equivalent if their maximum likelihood (ML) decoders coincide for every code. We show that this equivalence relation partitions the space of channels into a generalized hyperplane arrangement. With this, we define a coding distance between channels in terms of their ML-decoders which is meaningful from the decoding point of view, in the sense that the closer two channels are, t...

The Distance 4-Sector of Two Points Is Unique

The (distance) k-sector is a generalization of the concept of bisectors proposed by Asano, Matoušek and Tokuyama. We prove the uniqueness of the 4-sector of two points in the Euclidean plane. Despite the simplicity of the unique 4-sector (which consists of a line and two parabolas), our proof is quite non-trivial. We begin by establishing uniqueness in a small region of the plane, which we show...

The average distance of the n-th neighbour in a uniform distribution of random points

We first review the derivation of the exact expression for the average distance 〈rn〉 of the n-th neighbour of a reference point among a set of N random points distributed uniformly in a unit volume of a D-dimensional geometric space. Next we propose a ‘mean-field’ theory of 〈rn〉 and compare it with the exact result. The result of the ‘mean-field’ theory is found to agree with the exact expressi...