Some Results on Monotonicity of Volume and Surface Area of Objects in K Dimensions

Hypergeometry of objects in K dimensions is considered. In particular, the K dimensional sphere, polytope, cube, scaled polytope and the scaled cube are studied. The volume and the surface area of these objects are shown to reach a maximum for a particular value of dimension. The dimension corresponding to the maximum volume and to the maximum surface area is derived as a function of the radius. Furthermore, the p-norm in K dimensions is shown to be monotonically increasing in K, and monotonically decreasing in p.