Your minimization problem is equivalent to \begin{equation*} \min_{R^TR=I}\quad\prod_{i=1}^p r_i^T\Sigma r_i, \end{equation*} and it can be shown (using Hadamard's determinant inequality and some more argumentation) that this minimum overall $p$ orthonormal tuples is achieved by choosing the $r_i$ corresponding to the smallest $p$ eigenvectors.
Suvrit
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