Let $\Delta^n=\{(t_0,t_1,\dots,t_n)\in \mathbb{R}^{n+1} \mid \sum t_i=1,t_i\geq 0\}$ be the $n$-dimensional simplex. Let $X$ be the $k$-skelton of this simplex. What is the minimal m such that there exists a piecewise linear embedding $X\hookrightarrow \mathbb{R}^m$?
We can consider $m$ as a function with variables in $n,k$ and we call $m:=m(n,k)$. For example $m(3,1)=2$, $m(3,2)=3$. Since $H_k(X,\mathbb{Z})\neq 0$ and $H_{k+1}(X,\mathbb{Z})=0$, my guess is $m(n,k)=k+1$. If this is the case, is there a canonical way to embed $X$ into $\mathbb{R}^m$ piecewise linearly?