Introduction Dilations and Free Spectrahedral Inclusions Lifting and Averaging A Simplified Form for $\vartheta $ $\vartheta$ is the Optimal Bound The Optimality Condition $\alpha =\beta $ in Terms of Beta Functions Rank versus Size for the Matrix Cube Free Spectrahedral Inclusion Generalities Reformulation of the Optimization Problem Simmons' Theorem for Half Integers Bounds on the Median and the Equipoint of the Beta Distribution Proof of Theorem 2.1 Estimating $\vartheta (d)$ for Odd $d$. Dilations and Inclusions of Balls Probabilistic Theorems and Interpretations continued Bibliography Index.