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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.

### references

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.

4 added 114 characters in body; added 396 characters in body

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.

### references

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.

3 added 555 characters in body

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. 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.

### references

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.

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