Computations of entropy bounds: Multidimensional geometric methods

The entropy bounds for constructive upper bound on the needed number-of-bits for solving a dichotomy is represented by the quotient of two multidimensional solid volumes. For minimization of this upper bound exact calculation of the volume of this quotient is needed. Three methods for exact computing of the volume of a given nD volume are presented: (1) general method for calculation any nD volume by slicing it into volumes of decreasing dimension is presented; (2) a method applying appropriate curvilinear coordinate system is described for volume bounded by symmetrical curvilinear hypersurfaces (spheres, cones, hyperboloids, ellipsoids, cylinders, etc.); and (3) an algorithm for dividing any nD complex into simplices and computing of the volume of the simplices is presented, supplemented by a general formula for calculation of volume of an nD simplex. These mathematical methods enable exact calculation of volume of any complicated multidimensional solids. The methods allow for the calculation of the minimal volume and lead to tighter bounds on the needed number-of-bits