  We present an optimization model defined on the manifold of the set of
stochastic matrices. Geometrically, the model is akin to identifying a
maximum-volume $n$-dimensional simplex that has a given barycenter and is
enclosed by the $n$-dimensional standard simplex. Maximizing the volume of a
simplex is equivalent to maximizing the determinant of its corresponding
matrix. In our model, we employ trace maximization as a linear alternative to
determinant maximization. We identify the analytical form of a solution to this
model. We prove the solution is optimal and present necessary and sufficient
conditions for it to be the unique optimal solution. Additionally, we show the
identified optimal solution is an inverse $M$-matrix, and that its eigenvalues
are the same as its diagonal entries. We demonstrate how the model and its
solutions apply to the task of synthesizing conditional cumulative distribution
functions (CDFs) that, in tandem with a given discrete marginal distribution,
coherently preserve a given CDF.
