Name of set $\sum_{i=1}^nx_i\leq c, x_i\geq 0 \quad \forall i$.

Question

Fix $n$ as a positive integer. Fix $c>0$. The following set $A$ is called a "simplex": $$\mbox{$A = \left\{x\in\mathbb{R}^n: \sum_{i=1}^n x_i=c, x_i\geq 0 \quad\forall i \in \{1, ..., n\}\right\}$}$$ Is there a corresponding name for the following related set? $$ \mbox{$B = \left\{x\in\mathbb{R}^n: \sum_{i=1}^nx_i\leq c, x_i\geq 0 \quad\forall i \in \{1, ..., n\}\right\}$}$$ I want to call set $B$ a "subsimplex" but I wonder if there are alternative established names. I am using such sets in the context of convex optimization.

Answer 1

It's another simplex! (By definition, a simplex is a $k$-dimensional polytope that is the convex hull of its $k+1$ vertices.) If we use the names $A_n$ and $B_n$ to indicate the dimension of the ambient space, then $A_n$ is an $(n-1)$-dimensional simplex and $B_n$ is an $n$-dimensional simplex. Indeed, the map from $B_n$ to $A_{n+1}$ given by $$ (x_1,\dots,x_n) \mapsto \biggl(x_1,\dots,x_n,c-\sum_{i=i}^n x_i\biggr) $$ is an isomorphism. (Try this map out when $n=1$ and $n=2$ to gain some intuition.)