We show that for any convex body Q in the plane, the average distance from the Fermat-Weber center of Q to the points in Q is at least Δ(Q)/6, where Δ(Q) denotes the diameter of Q. Our proof is simple and straightforward, since it needs only elementary calculations. This simplifies a previously known proof that is based on Steiner symmetrizations.

Clustering methods are divided into hierarchical clustering, partitioning clustering, and more. K-Means is a method of partitioning clustering. We improve the performance of a K-Means, selecting the initial centers of a cluster through a calculation rather than using random selecting. This method maximizes the distance among the initial centers of clusters. Subsequently, the centers are distributed evenly and the results are more accurate than for initial cluster centers selected at random. This is time-consuming, but it can reduce the total clustering time by minimizing allocation and recalculation. Compared with the standard algorithm, F-Measure is more accurate by 5.1%.

A mesh spiral network (MSnet) and a mesh random (MRnet) are proposed. The MSnet consists of the 2-D torus and bypass links that keep the degree at six. The MRnet consists of the 2-D torus and random bypass links that keep the degree at six. The diameter and the average distance are calculated by using a computer program. The cost of the MSnet is slightly higher than that of the de Bruijn graph, and is about the same as the Star graph. The cost of the MRnet is better than that of the de Bruijn graph. The MSnet is proven to be maximally fault-tolerant. The upper bound of the MRnet size is also discussed.