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 beying 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 $s \in \{1,2... N\}$. 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?*