The convex hull of a finite set of affinely independent points; an n-simplex is the convex hull of n+1 affinely independent points and is the simplest possible n-dimensional polytope.

Principle

A simplex is the minimal convex building block in a given dimension: any point in the simplex has unique barycentric coordinates relative to its vertices, and every affine map is determined by its action on the vertices.

Demonstration

The standard n-simplex in R^{n+1} is {x in R^{n+1} : x_i >= 0, sum_i x_i = 1}. In low dimensions this gives a segment (1-simplex), triangle (2-simplex) and tetrahedron (3-simplex).

Misapplication

Calling the convex hull of n+1 points a simplex without checking affine independence (if dependent the hull is lower-dimensional), or confusing simplices with cells in a simplicial complex that may be glued non-affinely.

Consequence

Simplices provide barycentric coordinates, simple formulas for volume and orientation, and form the atoms of triangulations used to compute homology, perform integration, and approximate manifolds numerically.

Reversal

If the defining points are affinely dependent there is no n-simplex but a degenerate polytope of lower dimension; reversing the simplex idea yields general polytopes that require more vertices and faces.

Boundary

The notion presupposes an ambient affine space and affine independence; it excludes non-convex or curved analogues (e.g. geodesic simplices on curved spaces require additional structure) and combinatorial simplices without geometry.

Semantic Tension

Tension arises between the geometric simplex (an actual convex set with barycentric coordinates) and the purely combinatorial simplicial complex notion where a 'simplex' may denote an abstract set of vertices without embedding.

Synthesis

A simplex is the unique convex polytope determined by affinely independent vertices; as the elementary affine cell it underpins triangulations, coordinate representations, and local linear approximations of geometry.