Suppose we have abstract polytope $F$ of dimension $d$ (that is the greatest rank facet has rank $n$). Such abstract object may have realisations in d-dimensional Euclidean space as polytopes $A_i(F)$, and in some cases such realisation may have properly defined d-dimensional volume $V_i$ (with various values, depending on realisation $i$ shape).
Every such realisation defines realisation of abstract polytope $F_d$ which certainly defines further realisations of all it's facets $F_k$ for all ranks $k$ from maximal one $d$ down to $0$ .
Suppose we restrict our attention to such realisations that polytopes may be bounded within a sphere of finite volume $1.$ So the maximal volume of realisations is no bigger than $1.$ Further assume, that if particular realisation has no defined $d$-dimensional volume (because of missing face perhaps, or being non-orientable particular kind), we would set $V=0$.
We will work with Hasse diagrams of abstract polytope, as described for example here. Important for us would be two facts:
- Abstract polytopes with isomorphic Hasse diagrams are isomorphic
- Hasse diagrams are ordered decreasing, from top to bottom by rank of facets.
Suppose we'll assign to the greatest d-facet a number $M_d=\max (V_i)$, equal to the maximum volume of all realisations given abstract polytope $F_d$ (which is clearly finite and no bigger than volume of bounding sphere, that is 1). This is trivial case.
Suppose we don't know maximum volume $M_d$ but instead we have knowledge about such numbers for all N facets of $d-1$ kind, and it is equal to $M_{(d-1)}^s$, where $s \in \{1,2,\ldots, N\}$. Notice that $M_{(d-1)}^s$ are just (d-1) dimensional volumes of all facets. So we know $N$ numbers instead of one.
Q1: Is it possible to know from this information any bound on $M_d$ (that is is it possible to obtain any information on maximal volume among realisations $A_i(F)$)?
Let's further generalise: suppose we have no knowledge on any facets of rank bigger than n but we know all numbers $M_n^s$ where s numbers all n-dimensional facets of abstract polytope $F$.
Q2: Is it possible to know from this information any bound on $M_d$ (that is is it possible to obtain any information on maximal volume among realisations $A_i(F)$)?
In particular, has the question below known answer?
Q0: Is it possible to know any bounds on maximal volume of realisations of abstract polytope if we know the maximal amount of the length of all the edges (facets of rank $1$) among all its realisations?