Discrete Geometry: Info, Knowledge & More

[evinda]

Hello!
Can you give me information about the subject Discrete Geometry?
What is it about? What knowledge is required? (Thinking)

 

[caffeinemachine]

Hello!
Can you give me information about the subject Discrete Geometry?
What is it about? What knowledge is required? (Thinking)

To me discrete geometry means the subject of convex polytopes. Addtionally, one may want to read about properties of convex sets in general to gain a better understanding of polytopes but this I wouldn't count as discrete geometry per se.

As for the background, I think basic linear algebra and analysis would suffice. Some basic things about interior, closure, boundary of a topological space, compactness, closedness etc are needed. But I should mention that one does not need to know about these in the most abstract setting. In fact the topological notions are very intuitive in the setting of polytopes and convex sets so I would say that even this is not necessary.

As for books, check out Branko Grunbaum's Convex Polytopes.

 

[evinda]

Now I found that the following stuff will be done:

• Introduction in convex polytopes: examples of polytopes in different fields of mathematics, partial ordering of the sides, polarity, $f$- and $h$-vector, shellings, the theorem of upper bound of McMullen
• Graph of a polytope, Steinitz theorem, diameter of a graph and Hirsch conjecture
• Polytopes in problems of linear programming and optimization (Simplex method)
• Minkowski sum of polytopes, hyperplane arrangements , characteristic polynomial, Zaslavsky theorem
• Gale diagrams
• Rational polytopes, enumeration of integer points of a (rational) polytope, theorem of Ehrhart
• Polyhedral subdivisions and fiber polytopes
• Applications in [m] polymake [/m] and [m] Sagemath [/m].

Is the subject somehow related to theoretical computer science?

 

[caffeinemachine]

Now I found that the following stuff will be done:

• Introduction in convex polytopes: examples of polytopes in different fields of mathematics, partial ordering of the sides, polarity, $f$- and $h$-vector, shellings, the theorem of upper bound of McMullen
• Graph of a polytope, Steinitz theorem, diameter of a graph and Hirsch conjecture
• Polytopes in problems of linear programming and optimization (Simplex method)
• Minkowski sum of polytopes, hyperplane arrangements , characteristic polynomial, Zaslavsky theorem
• Gale diagrams
• Rational polytopes, enumeration of integer points of a (rational) polytope, theorem of Ehrhart
• Polyhedral subdivisions and fiber polytopes
• Applications in [m] polymake [/m] and [m] Sagemath [/m].

Is the subject somehow related to theoretical computer science?

Yes. This subject is related to convex and combinatorial optimization and linear programming which are studies by theoretical computer scientists.

 

[papiya]

Can anyone suggest some good books for discrete geometry for beginners? Or any useful websites or links.