The constructive bounds on the needed number-of-bits (entropy) for solving a dichotomy (i.e., classification of a given data-set into two distinct classes) can be represented by the quotient of two multidimensional solid volumes. Exact methods for the calculation of the volume of the solids lead to a tighter lower bound on the needed number-of-bits--than the ones previously known. Establishing such bounds is very important for engineering applications, as they can improve certain constructive neural learning algorithms, while also reducing the area of future VLSI implementations of neural networks. The paper will present an effective method for the exact calculation of the volume of any n-dimensional complex. The method uses a divide-and-conquer approach by: (i) partitioning (i.e., slicing) a complex into simplices; and (ii) computing the volumes of these simplices. The slicing of any complex into a sum of simplices always exists, but it is not unique. This non-uniqueness gives us the freedom to choose that specific partitioning which is convenient for a particular case. It will be shown that this optimal choice is related to the symmetries of the complex, and can significantly reduce the computations involved.