Convex geometry

Convex geometry is a part of mathematics that looks at special shapes and spaces called convex sets. These sets have a special rule: if you choose any two points inside the set, the straight line between them stays completely inside the set. This idea helps mathematicians solve many different kinds of problems.

Convex sets are found in many areas, such as computational geometry, where computers work on geometric problems, and linear programming, which helps find the best answer to tricky questions. They are also used in probability theory, which studies how likely events are, and game theory, the study of strategies and choices.

This part of geometry helps us solve real-world problems by giving clear rules about shapes and spaces. It links to many other areas of math, making it important for students and researchers.

Classification

Convex geometry is a part of math that studies shapes called convex sets. It has three main areas: general convexity, polytopes and polyhedra, and discrete geometry.

General convexity has many smaller topics. These include convex sets in different sizes, convex functions, and problems about measuring length, area, and volume. It also includes special types of convex sets and ideas about how they connect.

Historical note

Convex geometry is a new area of math. Early ideas about it appear in the works of ancient mathematicians like Euclid and Archimedes. It became its own subject in the late 1800s and early 1900s thanks to important work by Hermann Brunn and Hermann Minkowski. In 1934, T. Bonnesen and W. Fenchel wrote a big review of convex geometry in Euclidean space Rⁿ. The subject kept growing throughout the 20th century, connecting to many other areas of math.