Consider $n$ variables $a_1,...,a_n$ ranging over $\mathbb{R}^+$. Suppose we are given $n$ pairs of positive rational numbers $(p_1,q_1),...,(p_n,q_n)$ where each pair imposes bounds on the corresponding variable: $p_k\le a_k\le q_k$.
Now, suppose we constructed a polyhedron with maximal possible volume, having $n$ faces (indexed in some way) such that the area of $k$-th face lies within the given rational bounds for $a_k$. Let $V_{(p_1,q_1),...,(p_n,q_n)}$ denote the volume of the constructed polyhedron.
- Are polyhedra constructed this way always convex?
- Are $V_{(p_1,q_1),...,(p_n,q_n)}$ always algebraic numbers?
- If yes, can we narrow down this class of numbers to a smaller one?
- If no, can we describe in some simple way (without repeating definitions from this post) a countable class of real numbers (preferably, smallest one) such that $V_{(p_1,q_1),...,(p_n,q_n)}$ always belong to this class?
- Is there an algorithm that, given rational bounds $(p_1,q_1),...,(p_n,q_n)$ and a pair of rational numbers $(r, s)$ can decide if $r\le V_{(p_1,q_1),...,(p_n,q_n)}\le s$?
