A new definition of neighborhood of a point in multi-dimensional space

Given a set of points in multi-dimensional space, we propose a new definition for the neighbors of an arbitrary point P. The definition tries to capture the idea that the neighbors should be as near to P and as symmetrically placed around P as possible. In contrast, the conventional nearest neighborhood considers only nearness as the criterion for neighborhood. We propose an iterative procedure to compute the neighbors where the first neighbor is the nearest neighbor. The second and other neighbors are chosen so that at any stage the distance between the centroid of the neighbors and P is as small as possible. The centroid criterion takes care of symmetrical placement of the neighbors. One can use median instead of centroid to define the neighbors. The new definition is free from any user-specified parameter and can be used for pattern classification, clustering and low-level description of dot patterns.