It is conjectured that there exist at most 2k equidistant points in the k- dimensional rectilinear space. This conjecture has been verified for k 3; we show here its validity in dimension k = 4. We also discuss a number of related questions. For instance, what is the maximum number of equidistant points lying in the hyperplane: P k i=1 x i = 0 ? If this number would be equal to k, then the above conjecture would follow. We show, however, that this number is k + 1 for k 4. 1

Traditionally, an unsupervised classification divides all pixels within an image into a corresponding class pixel by pixel; the number of clusters usually needs to be fixed a priori by a human analyst. In general, the spectral properties of specific information classes change with the seasons, and therefore, the relation between object class and spectral cluster is not constant over time. In addition, relations for one image can in general not be extended to others. Thus, even if the number of clusters is correctly fixed for one image at one instance in time, the results cannot be transferred to other areas or epochs. In this study, a heuristic method based on Genetic Algorithms (GA) is adopted to automatically determine the number of cluster centroids during unsupervised classification. The optimization is based on the Davies-Bouldin Index (DBI). A software programme was developed in MATLAB,- and the GA unsupervised classifier was tested on an IKONOS satellite image. The classification results were compared to conventional ISODATA results, and to ground truth information derived from a topographic map for the estimation of classification accuracy. 1.