Convex polytope in the context of "Hypercube"

In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is one less than that of the ambient space. Two lower-dimensional examples of hyperplanes are one-dimensional lines in a plane and zero-dimensional points on a line.

Most commonly, the ambient space is n-dimensional Euclidean space, in which case the hyperplanes are the (n − 1)-dimensional "flats", each of which separates the space into two half spaces. A reflection across a hyperplane is a kind of motion (geometric transformation preserving distance between points), and the group of all motions is generated by the reflections. A convex polytope is the intersection of half-spaces.

In geometry, a tessellation of dimension 2 (a plane tiling) or higher, or a polytope of dimension 3 (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit. In other words, for any two faces A and B, there must be a symmetry of the entire figure by translations, rotations, and/or reflections that maps A onto B. For this reason, convex isohedral polyhedra are the shapes that will make fair dice.

Isohedral polyhedra are called isohedra. They can be described by their face configuration. An isohedron has an even number of faces.

In geometry, a tetrahedron (pl.: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices. The tetrahedron is the simplest of all the ordinary convex polyhedra.

The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex.

In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.

There are six convex and ten star regular 4-polytopes, giving a total of sixteen.

Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization).

More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. Its objective function is a real-valued affine (linear) function defined on this polytope. A linear programming algorithm finds a point in the polytope where this function has the largest (or smallest) value if such a point exists.

In geometry, a cross-polytope, hyperoctahedron, orthoplex, staurotope, or cocube is a regular, convex polytope that exists in n-dimensional Euclidean space. A 2-dimensional cross-polytope is a square, a 3-dimensional cross-polytope is a regular octahedron, and a 4-dimensional cross-polytope is a 16-cell. Its facets are simplexes of the previous dimension, while the cross-polytope's vertex figure is another cross-polytope from the previous dimension.

The vertices of a cross-polytope can be chosen as the unit vectors pointing along each co-ordinate axis – i.e. all the permutations of (±1, 0, 0, ..., 0). The cross-polytope is the convex hull of its vertices. The n-dimensional cross-polytope can also be defined as the closed unit ball (or, according to some authors, its boundary) in the ℓ₁-norm on R, those points x = (x₁, x₂..., x_n) satisfying