Convex polytope

A convex polytope is a special shape that belongs to a larger group of shapes called polytopes. What makes a convex polytope unique is that it is also a convex set, meaning that if you pick any two points inside the shape, the straight line connecting them will always stay inside the shape. Convex polytopes can exist in different dimensions — for example, a polygon in 2D (like a square or triangle) or a polyhedron in 3D (like a cube or pyramid) can be convex polytopes.

These shapes are very important in many areas of mathematics. They help mathematicians understand how shapes behave and how they can be broken down into simpler parts. Beyond pure math, convex polytopes are also used in practical problems, especially in something called linear programming. This is a way to find the best solution to a problem with many choices and limits, like figuring out the most efficient way to use resources.

Even though the idea of convex polytopes might sound complex, they are really just nice, well-behaved shapes that help us solve many kinds of problems — both in theory and in real life.

Terminology

In geometry, a polytope is a special shape that can be bounded or unbounded. Many people use the word "polytope" to mean a bounded convex polytope, while others use "polyhedron" for shapes that might not be bounded. Some writers, like Grünbaum and Ziegler, simply call convex polytopes "polytopes" to avoid saying "convex" too many times.

A polytope is called full-dimensional if it takes up all the space in an n-dimensional world, like a 3D shape living completely in our 3D space.

Examples

Many examples of bounded convex polytopes are found in convex polyhedra and convex polygons.

In two dimensions, unbounded convex polytopes include a half-plane, a strip between two parallel lines, an angle shape formed by two non-parallel half-planes, and a shape made from a convex polygonal chain with two rays attached to its ends. In higher dimensions, examples include a slab between two parallel hyperplanes, a wedge from two non-parallel half-spaces, a polyhedral cylinder (an infinite prism), and a polyhedral cone (an infinite cone) defined by three or more half-spaces meeting at a common point.

Definitions

A convex polytope is a special type of shape in space. It is both convex, meaning any line between two points inside it stays entirely inside, and bounded, meaning it doesn’t stretch out forever. Convex polytopes are important in many areas of math and applications like optimization.

Convex polytopes can be described in two main ways. One way is by listing a finite set of points called vertices; the polytope is the smallest convex shape that includes all these points. This is called the vertex representation. Another way is by stating a set of linear inequalities; the polytope is the set of all points that satisfy these inequalities. This is called the half-space representation. Both descriptions help us understand and work with these shapes in different ways.

Properties

Every convex polytope is made from points that are a special mix of its corners, called a convex combination. Unlike other shapes, these polytopes have special flat surfaces called faces, edges, vertices, and ridges.

A face is where the polytope meets a flat space without including any inside points on its edge. The collection of all these faces forms a structure called a face lattice. This lattice helps us understand how different polytopes can be the same in shape but different in size or position.

Algorithmic problems for a convex polytope

Different ways to describe a convex polytope can be useful for different tasks, so turning one description into another is an important problem. For example, finding all the corners of a polytope is called the vertex enumeration problem, while finding all its sides is called the facet enumeration problem. Special computer programs called convex hull algorithms can help with these tasks.

One important task is to calculate the amount of space a convex polytope takes up, known as its volume. This can be done roughly using methods like convex volume approximation, especially if we can check whether a point is inside the polytope. However, finding the exact volume can be tricky because the numbers involved might get very large compared to the size of the description of the polytope.