Linear combination of norms in improving approximation of Euclidean norm

In the past, different distance functions and their combinations had been proposed as good approximators of Euclidean metrics. In particular in recent years, a few distance functions with their general forms in n-dimensional real and integer spaces were identified for their improved performances in approximating corresponding Euclidean metrics. In this paper, we have identified a linear combination of two such distance functions from the families of weighted distances (WD) and weighted t-cost distances (WtD), which provides significant improvement over the past results in the quality of approximation. Further, we discuss a special case of linear combination, convex combination of distances, and provide optimal combinations by minimizing mean square error (MSE). In this case also the proposed pair of norms perform superior to other reported combinations. In our study, we also present new results related to characterization of overestimated and underestimated norms of Euclidean norm by their hyperspheres. The analysis leads to new results on theoretical bounds of maximum relative error (MRE) of some of the existing distance functions, including their linear combinations.