Fix a positive integer $n$. For any convex polytope $\mathcal{P}$ of dimension $n$, let $V_\mathcal{P}$ denote its volume and let $K_{\mathcal{P}}$ denote the maximum possible volume of an $n$-dimensional hyperrectangle completely contained in $\mathcal{P}$.
What is the minimum possible value of the ratio $$\frac {K_{\mathcal{P}}}{V_{\mathcal{P}}}$$ taken over all bounded convex polytopes $\mathcal{P}$ of dimension $n$?
If we cannot compute the minimum value of the ratio exactly, can we prove some lower bounds for this quantity?
Note: The application I am interested involves a much more constrained version of the above problem. This alternate version is written below, and any progress on that question is greatly appreciated as well.
Alternate Problem
The problem is the same as above, except we consider $\mathcal{P}$ living in $\mathbb{R}^n$, and the faces of $\mathcal{P}$ are all of the form $x_i = c_i$ or of the form $x_j - x_i = c_{ij}$, where the $c_i$ and $c_{ij}$s are constants. In other words, the faces are either perpendicular to the axes, or are very speciffic faces that cut two of the faces at $45^{\circ}$.
Moreover, $K_{\mathcal{P}}'$ is taken to be the maximum possible value volume of a hyperrectangle with sides parallel to the axes in $\mathbb{R}^n$ that is contained in $\mathcal{P}$.
In this modified scenario, we still want to compute lower bounds on $$\min \frac {K_{\mathcal{P}}'}{V_{\mathcal{P}}}$$ where the minimum is now taken taken over all bounded convex polytopes $\mathcal{P}$ of dimension $n$, with the constraints on types of faces?