Let $\mathbf{A}=[a_{ij}]$ be a matrix of real numbers $a_{ij}>0$ for all $i\in\{1,\ldots,k\}$ and $j\in\{1,\ldots,n\}$ and let $v$ be a nonnegative real number. The coefficient of the matrix are not *all* greater than $1$ nor *all* less than one.

The problem is to find a subset $S$ of the columns (of $\{1,\ldots,n\}$) where $S\neq \emptyset$ and $S$ is not a singleton such that $$\delta(S, \mathbf{S}) = \dfrac{\prod_{j\in S} ||b_j||}{\sqrt{\det(\mathbf{S}^\top \mathbf{S})}}\leqslant v.$$ Here $\mathbf{S}$ is a matrix whose columns are $b_j:=[a_{1j}, a_{2j}, \ldots, a_{kj}]^\top$ for $j\in S$.

In fact, in the optimization settings, I would like to find a subset $S$ of vectors which are as nearly orthogonal as possible. Or simply $$\underset{S\subseteq \{1,\ldots,n\}\\S\neq\emptyset}{\text{minimize }\;} \delta(S, \mathbf{S}).$$

*Example*: Take $v=0.17$ and $$\mathbf{A}=\begin{bmatrix} 0.1&2&3\\2&0.2&0.1\end{bmatrix}.$$ In $S=\{1, 2\}$, the matrix is $$\mathbf{S}=\begin{bmatrix} 0.1&2\\2&0.2\end{bmatrix}.$$

This example has the solution $S=\{1,3\}$. Its value $\delta(S, \mathbf{S})=0.1675$.

How to solve this problem? Is this problem known (or is there any similar problem in the literature)?

Besides enumerating all subsets of $\{1,\ldots,n\}$, I cannot find a good way to solve the problem.

**EDIT**

According to Federico's comment, the problem seems to be equivalent to $$\underset{S\subseteq \{1,\ldots,n\}\\S\neq\emptyset}{\text{maximize }\;} \det(\mathbf{S}^\top\mathbf{S}),$$ where $\mathbf{S}$ is the matrix whose columns are $[a_{1j}, a_{2j}, \ldots, a_{kj}]^\top$ for $j\in S$.