# Polytope (geometry)

Dragging takes you from the point, to the straight line, to the surface, to the 3D cube and the 4D cube

A polytope ( that , from ancient Greek πολύς polýs 'much' and τόπος tópos 'place'; plural polytopes ) in geometry is a generalized polygon in any dimension. One speaks of -Polytopes, where the dimension is. ${\ displaystyle d}$${\ displaystyle d}$

## definition

A 0 polytope is a single corner (a point); a 1-polytope consists of two corners connected by an edge; a 2-polytope consists of several 1-polytopes, each connected at a corner, forming a cycle and thus represents a polygon ; a 3-polytope in turn consists of several 2-polytopes connected at the edges and thus represents a polyhedron ; etc.

In general, a polytope is formed from several polytopes, each of which can have one subpolytope in common with one another (such as the common corner of two edges or the common edge of two surfaces). All sub-polytopes must be contained in exactly two poly- polytopes, which are then considered to be adjacent. Furthermore, a chain of neighboring polytopes must exist between two polytopes , so that two links are connected in the manner described by a subpolytope - for example, several disjoint polygons do not form a 3-polytope. ${\ displaystyle (d \! + \! 1)}$${\ displaystyle d}$${\ displaystyle (d \! \! - \! \! 1)}$${\ displaystyle (d \! \! - \! \! 1)}$${\ displaystyle d}$${\ displaystyle d}$${\ displaystyle d}$${\ displaystyle (d \! \! - \! \! 1)}$

## nomenclature

In certain dimensions, polytopes have been given special names, as listed in the following table:

dimension Name of the d polytop
0 Point
1 route
2 Polygon
3 polyhedron
4th Polychoir

If you consider a polytope of dimension d , the following terms exist:

dimension Name of the sub-poly top
0 corner
1 Edge
d - 3 Engl .: peak (roughly: "top")
d - 2 Ridge (e.g. corner of a polygon ( d = 2 ), edge of a tetrahedron ( d = 3 ), ...)
d - 1 Facet (e.g. edge of a polygon ( d = 2 ), side surface of a cube ( d = 3 ), ...)
d Engl .: body (roughly: "trunk")

The dimension of a polytope is defined as the dimension of its affine shell , i.e. the smallest affine space that it contains. So a cube is three-dimensional because the smallest space it contains is three-dimensional. A real polytope is a polytope that does not lie entirely in a real subspace, i.e. has the same dimension as the space under consideration. ${\ displaystyle P}$${\ displaystyle P}$

## Convex polytopes

Special meaning in mathematics and linear programming have convex polytopes (often just polytope ), ie, so that the link between any two points of the polytope in turn is contained completely in the polytope polytopes. These are exactly the constrained convex polyhedra . Equivalently, they can be defined as the convex hull of a finite number of points (such as the corner points).

Each actual polytope divides space into its interior, its exterior and its edge. Every segment that connects an inner and an outer point intersects the edge in exactly one point. The intersection of two actual polytopes with a common inner point is again an actual polytope. By induction the same follows for a finite number of proper polytopes with a common internal point.

Each facet (end point for lines, edge for polygons, etc.) of a polytope can be assigned a half-space on the edge of which the facet lies and which contains the polytope. To do this, imagine the part of the space that lies on the side of the side surface facing the polytope. Such a half-space can be understood as the set of points that satisfy a linear inequality in their Cartesian coordinates . The intersection of all the half-spaces to each of the facets is in turn the polytope. Thus every convex polytope can be understood as a solution set of a linear system of inequalities in finitely many variables. Insofar as the solution set of a linear system of inequalities is restricted (i.e. the distance between all points is restricted), the converse also applies.

Is

${\ displaystyle a ^ {T} x \ leq b}$

a linear inequality which is satisfied by all points of the polytope, then the intersection of the polytope with the set

${\ displaystyle \ {x | a ^ {T} x = b \}}$

referred to as the side face. Each side surface can be represented by such an inequality. In the special case of the inequality

${\ displaystyle 0 ^ {T} x \ leq 0}$

the whole polytope results as the intersection, and for the inequality

${\ displaystyle 0 ^ {T} x \ leq 1}$

is the cut

${\ displaystyle \ {x | 0 ^ {T} x = 1 \} \ cap P = \ {x | 0 ^ {T} x = 1 \} = \ varnothing}$

the empty crowd. The set of all faces of a polytope is respect. Inclusion association ordered . A facet of a -dimensional convex polytope is then a -dimensional side surface. In the case of a three-dimensional cube, for example, all corners, edges and faces of the cube are “side faces”, but also the empty set and the entire cube. But only the two-dimensional side surfaces are facets of the cube. ${\ displaystyle n}$${\ displaystyle (n-1)}$

A corner of a convex polytope is a point in the polytope that cannot be convexly combined by other points of the polytope, i.e. it does not lie on a line between two other points of the polytope. This corresponds to the graphic idea of ​​a corner. For example, you cannot construct a line between two points on a cube that contains a corner as an inner point. A corner of a polytope is said to be degenerate if the number of facets it contains is greater than the dimension of . For example, the tip of a three-dimensional pyramid with a square base is degenerate because it is contained in four facets. A convex polytope is called integer if all of its corners are described by integer coordinates. Among other things, these terms are important in linear and integral linear optimization , because the optimum of a linear program is always assumed in one corner. ${\ displaystyle x}$${\ displaystyle P}$${\ displaystyle x}$${\ displaystyle P}$