Convex geometry

Convex geometry is a fascinating part of mathematics that explores special shapes and spaces called convex sets. These sets have an important property: if you pick any two points inside the set, the straight line connecting them stays entirely within the set. This idea helps mathematicians understand many kinds of problems in different fields.

Convex sets appear naturally in many areas of study, such as computational geometry, where computers solve geometric problems, and linear programming, which is used to find the best solution to complex questions. They are also important in probability theory, which looks at how likely events are, and game theory, the study of strategies and decisions.

This area of geometry helps us solve real-world problems by providing clear rules about shapes and spaces. It connects to many other parts of math, making it a key topic for students and researchers alike.

Classification

Convex geometry is a part of math that studies shapes called convex sets. According to the Mathematics Subject Classification, it has three main areas: general convexity, polytopes and polyhedra, and discrete geometry.

General convexity can be divided into many smaller topics, such as convex sets in different dimensions, convex functions, and problems about measuring length, area, and volume. It also includes special types of convex sets and theories about how they relate to each other.

Historical note

Convex geometry is a young area of math. Early ideas about it appear in the works of ancient mathematicians like Euclid and Archimedes. It grew into 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.