This monograph sets forth the results of a study of the geometry of a simplex, multiply-cut by hyperplanes, and its implications for constrained optimization. First, the study presents basic cases about the polytope which results when a simplex is cut by either one or two hyperplanes, having designated feasible and infeasible sides. Proceeding, the study examines the geometry of a polytope after multiple cuts of a simplex, and specifies a test to determine whether or not a set of cuts each deleting one vertex of a simplex, together delete the entire simplex. As well, the study provides an efficient algorithm for determining the status — uncut, partially cut, or completely cut — for any sub-simplex of the original simplex. Next, the study addresses the vertex and edge counts of a polytope resulting from cutting a simplex. At first, two cuts operate, and either they intersect within the simplex, or they do not. In the former case, the number of vertices and edges is a simple consequence of an earlier result. In the latter case, the incremental number emerges in the presentation. An extension to the many cut scenario ensues, with pairwise intersecting allowed within a simplex to provide a closed formula for vertices and edges. Following, the work discusses the `beta problem,' an optimization problem involving a feasible simplex constrained by inequalities among disjoint sets of the variables. For added interest, coefficients of the defining matrix are considered uncertain, and thus subject to probability distributions. The gist of this development comprises these and related ideas, with examples. As one commonly applies the simplex method to solve such `beta problems,' he necessarily considers quotient distributions on the variables appearing in the pivot equations. For reference, therefore, the study attaches an appendix which provides statistics for a few of the more commonly encountered quotient distributions.