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.

**EDIT** Here the details that complete my answer (because it seems that the OP already knew the answer and still posted this question...)

We know that for a matrix $\Sigma$ with eigenvalues $\lambda_1,\ldots,\lambda_n$, and any orthonormal $R$,

\begin{equation*}
  \prod_{i=n-p+1}^n \lambda_i \le \det(R^T\Sigma R) \le \prod_{i=1}^p (R^T\Sigma R)_{ii} = \prod_{i=1}^p r_i^T\Sigma r_i.
\end{equation*}
We are trying to minimize the upper, bound and by choosing $R$ to be the matrix of the smallest $p$ eigenvectors, we can actually turn the first inequality into an equality. This proves the claim. The first inequality is a classic result on eigenvalues, while the second inequality is *Hadamard's determinant theorem*.