I am looking for a hyperrectangle with maximum volume which is contained in an ellipsoid. My Ellipsoid is defined as $\mathcal{E}=\{x\in\mathbb{R}^n\mid x^T P x \leq c \}$, where $P>0$.
For smaller dimension I can write an optimization problem. But for $n=23$, the optimizer have to check $2^{23}$ vertices. Does somebody have a closed form solution?
Edit: I am looking for a hyperrectangle whose edges are parallel to the coordinate axes. And $P$ is not a diagonal matrix.
By diagonalizing $P$, your problem is equivalent to maximizing $\prod_{i=1}^n x_i$ subject to $\sum_{i=1}^n c_i x_i = 1$, $x_i \ge 0$ (the corresponding hyperrectangle having vertices $(\pm \sqrt{x_1}, \ldots, \pm \sqrt{x_n})$). Show (using a Lagrange multiplier if you want) that the optimal solution will have all $c_i x_i$ equal.