Simplex

Remarks

The numbers $t_0, \ldots, t_n$ are the barycentric coordinates of a point of $\Delta$, and they are unique by affine independence.

Example: $v_0 = (0, 0)$, $v_1 = (1, 0)$ and $v_2 = (0, 1)$ are affinely independent, since $v_1 - v_0$ and $v_2 - v_0$ are linearly independent. Their $2$-simplex is the triangle $\set{(x, y) : x, y \ge 0, \ x + y \le 1}$.

Non-example: the points $(0, 0)$, $(1, 0)$ and $(2, 0)$ are not affinely independent. Their convex hull is the segment from $(0, 0)$ to $(2, 0)$, a $1$-simplex but not a $2$-simplex.

The standard $n$-simplex is the simplex in $\R^{n+1}$ with the standard basis vectors as vertices. Some texts, and Mathlib, use the word simplex for the family of vertices rather than for their convex hull.