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Exactly what set of mathematical tools (means: set of areas of mathematical knowledge) are appropriate to begin with to analyse (to enumerate face vectors associated with polyhedron, to calculate the combinatorial types) the Convex Polyhedra (starting with simple polyhedron to general).

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-1: This question is prohibitively vague. What kind of "analysis" do you want to do? Your tag suggests that you want to somehow enumerate them, but the set of convex polyhedra is uncountable. –  Pete L. Clark Jan 24 '10 at 12:53
Yes, Pete is right about its vagueness; my pupose is to count face vectors. –  Ali Dino Jumani Jan 24 '10 at 13:19
Assuming you have 3-dimensional convex polyhedra in mind look at Branko Grunbaum's book Convex Polytopes. For k-valent polyhedra (k = 3, 5, or 5), if I understand what you want, then there are some partial results going under the name of what are called "Eberhard Theorems." –  Joseph Malkevitch Jan 24 '10 at 14:12
Joseph Malkevitch's own AMS monthly column "Euler's Polyhedral Formula: PartII" is also a nice elaboration in this respect; but naming the explicit set of tools will be helpful. –  Ali Dino Jumani Jan 24 '10 at 14:51
-1. I am not sure that "finding the exact set of mathematical tools" is a good way to learn things: that isn't how mathematics works. I mean, maybe you should learn homology theory, but maybe you don't need to bother... There are several books on convex polytopes and polyhedra, so if a list of such books is what you were after, perhaps you should edit the question? –  Yemon Choi Jan 24 '10 at 22:02
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2 Answers

Dear Ali,

Well there are various tools which are useful to the study of convex polytopes. The following list is perhaps not complete and it certainly should not be frightening. (I dont know very well various of these tools.)

1) Basic tools of linear algebra and convexity.

The notions of supporting hyperplanes, seperation theorems, Caratheodory, Helly and Radon theorem etc.

2) Combinatorics

Some of the study of convex polytopes translates geometric questions to purely combinatorial questions. So familiarity with combinatorial notions and techniques is useful.

3) Graph theory

As Joe mentioned the study of polytopes in 3 dimensions is closely related to the study of planar graphs. There are few other connections to graph theory so it is useful to be familiar with some graph theory.

4) Gale duality

The notion of Gale duality is a linear-algebra concept which privides an important technique in the study of convex polytopes.

5) Some basic algebraic topology

Euler's theorem and its higher dimensional analogues is of central imoprtance and this theorem is closely related to algebraic topology. Another example: there is a result by Perles that the [d/2]-dimensional skeleton of a simplicial polytope determines the entire combinatorial structure. (See this paper by Jerome Dancis.) The proof is based on an elementary topological argument. Borsuk-Ulam theorem also has various nice applications for the study of polytopes.

6) Some functional analysis

There is a result by Figiel, Lindenstrauss, and Milman that a centrally symmetric convex polytope in d dimension satisfies $$log f_0(P) \cdot log f_{d-1}(P) \ge \gamma d$$ for some absolute positive constant $\gamma$. The proof is based on a certain variation of Dvoretzky theorem and I am not aware of an alternative approach.

7) Some commutative algebra

Several notions and results from commutative algebra plays a role in the study of convex polytopes and related objects. Especially important is the notion of Cohen Macaulay rings and results about these rings.

8) Toric varieties

Understanding the topology of certain verieties called "toric varieties" turned out to be quite important for the study of convex polytopes.

All these items refer to general polytopes. There is also a (related) reach study of polytopes arising in combinatorial optimization. Here is a link to a paper entitled "Polyhedral combinatorics an annotated bibliography" by Karen Aardal and Robert Weismantel.


Here are some relevant references: Ziegler's book: Lectures on Convex Polytopes, and the second edition of Grunbaum's book "Convex polytopes" will give a very nice introduction to topics 1) - 4). The connection with commutative algebra and some comments and references to the connection with toric varieties (topics 7 and 8) can be found in (chapters 2 and 3 of) the second edition of Stanley's book "Combinatorics and Commutative Algebra". Relations with algebraic topology and with functional analysis can be found in various papers. This wikipedia article can also be useful.

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Wouldn´t the more interesting question be, really, what tools are useless to study polytopes? :) –  Mariano Suárez-Alvarez Sep 2 '10 at 12:45
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If you are looking for software, I recommend POLYMAKE, an open source program which can take a polyhedron specified either by inequalities or vertices and return (among many things) the $f$-vector.

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I am writing the terms "Explicit set of Tools" or the " Mathematical Tools" ; which means the mathematical machinary needed to enumerate face vectors associated with polyhedron, to calculate the combinatorial types. –  Ali Dino Jumani Jan 24 '10 at 18:27
Since planar 3-connected graphs correspond to 3-polytopal graphs by Steinitz's Theorem one typically starts with an equation, say (), derived from Euler's polyhedral formula.This formula can involve only numbers of faces or it can involve numbers of faces and vertices. If one successfully constructs a polyhedron whose values satisfy the equation () one tries to find another solution that satisfies (*) by using graph theory ideas to make some "local change" in the graph that solved the original case. Perhaps if you are more specific about the problem that interests I can be more specific. –  Joseph Malkevitch Jan 24 '10 at 19:22
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