They are a family of [Polyhedron] . The simplex determined by \(n\) vectors which have [Affine Independence] is given by the [Convex Hull] of all the vectors.
It can be written as \(C = \{ \theta_0 v_0 + \theta_1 v_1 + \dots + \theta_k v_k \mid \theta \succ 0, 1^T\theta = 1\}\) where \(\theta = \begin{bmatrix} \theta _1, \theta_2, \dots, \theta_k \end{bmatrix}^T\) and \(1\) is a vector of all ones.