Let $n \in \{1,2,\dots\}$.
Consider the convex set consisting of all $(x_1,\dots,x_n) \in \mathbb{R}^n$ satisfying:
- $x_i \geq 0$, $i \in \{1,\dots,n\}$
- $x_1 + \cdots + x_n \leq 1$
Does this set have a name?
More generally, suppose $\mathbf{V}$ is an $n$-dimensional, real vector space. Let $v_1, \dots, v_n$ be a basis for $\mathbf{V}$, and consider the convex set consisting of all $v \in \mathbf{V}$ of the form $v = x_1v_1 + \cdots + x_nv_n$, where the coefficients $x_i \in \mathbb{R}$ satisfy the two conditions listed above.
Does this set have a name?