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:

  1. Abstract polytopes with isomorphic Hasse diagrams are isomorphic
  2. 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?

  • 2
    $\begingroup$ Isoperimetric inequality gives such bounds. $\endgroup$ – Alexandre Eremenko May 27 '18 at 11:46
  • $\begingroup$ Looks good. Though as generalised polytope realisation may be naturally considered as polyhedra, I suppose in particular cases there may be more strict bounds, may not? $\endgroup$ – kakaz May 27 '18 at 12:34
  • $\begingroup$ And specially for relationship between rank 1 facets length ( edges) and volume of the greatest facet, I expect some tighter bounds. Basically in isoperimetric inequality there is volume of surrounding sphere mentioned, but in a case of rank 1 versus volume in d dimensional space, there is a lot of inequalities in between, which gives hope... $\endgroup$ – kakaz May 27 '18 at 13:41

When $M_{d-1}^s$ is the facet volume of the $s$-th facet, then it would contribute the largest part to the unknown polytope volume $M_d$ if it would be assumed to be a $(d-1)$-dimensional hyperball volume and thus contribute to $M_d$ as the according cone. That is, the volume of this cone, $V_d^{cone(s)}$, would provide an upper bound for the respective facet pyramid. (Here using the center of the given $d$-dimensional unit hyperball for the according tip.)

For that aim you'd have to use the volume formula of the general hyperball $$V_{d-1}^{ball}(R)=\frac{\sqrt{\pi^{d-1}}}{\Gamma(\frac{d+1}2)} R^{d-1}$$ and equate that to $M_{d-1}^s$, then solve it for $R=R_s$, resulting in $$R_s=\frac{\left(\Gamma(\frac{d+1}2)\ M_{d-1}^s\right)^{\frac1{d-1}}}{\sqrt{\pi}}$$ Now calculate the height of that cone according to Pythagoras to $$H_s=\sqrt{1-R_s^2}=\sqrt{1-\frac{(\Gamma(\frac{d+1}2)\ M_{d-1}^s)^{\frac2{d-1}}}{\pi}}$$ Thus finally the searched for upper bound for the facet pyramid would be the volume of the according cone. That one could be given as $$V_d^{cone(s)}=\frac1{d-1}\ M_{d-1}^s\ H_s=\frac{M_{d-1}^s}{d-1}\ \sqrt{1-\frac{(\Gamma(\frac{d+1}2)\ M_{d-1}^s)^{\frac2{d-1}}}{\pi}}$$ Thus you get your searched for bound as $$M_d \le \sum_s V_d^{cone(s)}$$ --- rk


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