simplex

A simplex is the simplest possible polytope in a given dimension, defined as the convex hull of k+1 affinely independent points in a Euclidean space. Its dimension is k. Examples include a 0-simplex being a point, a 1-simplex a line segment, a 2-simplex a triangle, and a 3-simplex a tetrahedron.

The standard k-simplex, denoted Δ^k, is the set of points (t0, ..., tk) in R^{k+1} with ti ≥ 0 for all i and sum ti = 1. Its vertices are the unit vectors e0, ..., ek. Any k-simplex is an affine image of Δ^k.

Faces of a simplex are obtained by taking convex hulls of subsets of its vertices. The convex hull of a subset of the k+1 vertices is a face whose dimension equals the size of the subset minus one. A point inside a simplex can be written uniquely as a convex combination p = sum λ_i v_i with λ_i ≥ 0 and sum λ_i = 1; the coefficients λ_i are called barycentric coordinates. The k-volume of a simplex can be computed by a determinant formula based on its vertex coordinates.

In topology and combinatorics, simplices are the building blocks of simplicial complexes, which assemble simplices along shared faces to model spaces. The standard simplex also serves as a model in algebraic topology for defining chains and homology.

Applications include mesh generation in computer graphics and finite element methods, as well as theoretical methods in linear programming, where the simplex method navigates between basic feasible solutions on the edges of a polyhedron.