On the Omnidimensional Convex Polytopes and n-Balls in Negative, Fractional and Complex Dimensions

The study shows that the recurrence relations defining volumes and surfaces of omnidimensional convex polytopes and n-balls are continuous and defined for complex n, whereas in the indefinite points their values are given in the sense of a limit of a function. The volume of an n-simplex is a bivalued function for n < 0, and thus the surfaces of n-simplices and n-orthoplices are also bivalued functions for n < 1. Applications of these formulas to the omnidimensional polytopes inscribed in and circumscribed about n-balls reveal previously unknown properties of these geometric objects in negative, real dimensions. In particular for 0 < n < 1 the volumes of the omnidimensional polytopes are larger than the volumes of circumscribing n-balls, while their volumes and surfaces are smaller than the volumes of inscribed n-balls. Specific products and quotients of volumes and surfaces of the omnidimensional polytopes and n-balls are shown to be independent of the gamma function.