Abstract

Dr Cai Wen defined in his 1983 paper - the distance formula between a point x0 and a one-dimensional 1D interval a b - and the dependence function which gives the degree of dependence of a point with respect to a pair of included 1D - intervals This paper inspired us to generalize the Extension Set to two-dimensions i e in plane of real numbers R2 where one has a rectangle instead of a segment of line determined by two arbitrary points A a1 a2 and B b1 b2 And similarly in R3 where one has a prism determined by two arbitrary points A a1 a2 a3 and B b1 b2 b3 We geometrically define the linear and non-linear distance between a point and the 2Dand 3D-extension set and the dependent function for a nest of two included 2D - and 3D - extension sets Linearly and non-linearly attraction point principles towards the optimal point are presented as well The same procedure can be then used considering instead of a rectangle any bounded 2D-surface and similarly any bounded 3D - solid and any bounded n D - body in Rn These generalizations are very important since the Extension Set is generalized from one-dimension to 2 3 and even n-dimensions therefore more classes of applications will result in consequence