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.evinda said:Hello!
Can you give me information about the subject Discrete Geometry?
What is it about? What knowledge is required? (Thinking)
Yes. This subject is related to convex and combinatorial optimization and linear programming which are studies by theoretical computer scientists.evinda said: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?
Discrete geometry is a branch of mathematics that deals with the study of geometric properties and structures that are considered “discrete,” meaning they have a finite or countable number of elements. It involves the use of combinatorics, graph theory, and other discrete mathematical tools to study geometric shapes and their properties.
Discrete geometry has numerous practical applications in fields such as computer science, physics, chemistry, and engineering. It is used in computer graphics and animation, pattern recognition, image processing, and computer vision. It also plays a role in designing efficient networks, coding theory, and molecular structure analysis.
Discrete geometry deals with geometric structures that are made up of distinct, separate elements, while continuous geometry deals with smooth, continuous geometric objects. In other words, discrete geometry focuses on finite or countable sets of points, lines, and shapes, while continuous geometry deals with infinite sets and smooth curves and surfaces.
Discrete geometry plays a crucial role in data analysis and data science by providing tools and methods for representing and analyzing data sets with complex geometric structures. It enables the visualization and manipulation of high-dimensional data, which is essential for understanding patterns and relationships in large datasets.
Yes, there are still many open problems and unsolved questions in discrete geometry, particularly in the areas of combinatorial geometry, discrete differential geometry, and geometric graph theory. These include questions about the existence and properties of certain geometric structures, as well as questions about the complexity and efficiency of algorithms for solving geometric problems.