A generalization of the Minkowski distance and a new definition of the ellipse

In this paper, we generalize the Minkowski distance by defining a new distance function in n-dimensional space, and we show that this function determines also a metric family as the Minkowski distance. Then, we consider three special cases of this family, which generalize the taxicab, Euclidean and maximum metrics respectively, and finally we determine circles of them with their some properties in the real plane. While we determine some properties of circles of the generalized Minkowski distance, we also discover a new definition for the ellipse.