Definition Points v₀, v₁, . . . , v_q in some Euclidean space R^k are said to be geometrically independent (or affine independent) if the only solution of the linear system

is the trivial solution λ₀ = λ₁ = · · · = λ_q = 0.

It is straightforward to verify that v₀, v₁, . . . , v_q are geometrically independent if and only if the vectors v₁ − v₀, v₂ − v₀, . . . , v_q − v₀ are linearly independent. It follows from this that any set of geometrically independent points in R^k has at most k + 1 elements. Note also that if a set consists of geometrically independent points in R^k, then so does every subset of that set.

Definition A q-simplex in R^k is defined to be a set of the form

where v₀, v₁, . . . , v_q are geometrically independent points of R^k. The points v₀, v₁, . . . , v_q are referred to as the vertices of the simplex. The non-negative integer q is referred to as the dimension of the simplex.

Note that a 0-simplex in R^k is a single point of R^k, a 1-simplex in R^k is a line segment in R^k , a 2-simplex is a triangle, and a 3-simplex is a tetrahedron.

Let σ be a q-simplex in R^k with vertices v₀, v₁, . . . , v_q. If x is a point of σ then there exist real numbers t₀, t₁, . . . , t_q such that

Moreover t₀, t₁, . . . , t_q are uniquely determined:

Hence t_j − s_j = 0 for all j, since v₀, v₁, . . . , v_q are geometrically independent. We refer to t₀, t₁, . . . , t_q as the barycentric coordinates of the point x of σ.

Lemma 1.1 Let q be a non-negative integer, let σ be a q-simplex in R^m, and let τ be a q-simplex in Rⁿ, where m ≥ q and n ≥ q. Then σ and τ are homeomorphic.

Proof Let v₀, v₁, . . . , v_q be the vertices of σ, and let w₀, w₁, . . . , w_q be the vertices of τ . The required homeomorphism h: σ → τ is given by

for all t₀, t₁, . . . , t_q satisfying 0 ≤ t_j ≤ 1 for j = 0, 1, . . . , q and