Can someone help me prove this;
I have managed to sketch them but cannot prove why its convex
Consider $\displaystyle x=(x_i),\,y=(y_i)$ in $\displaystyle S_{n-1}$ and $\displaystyle t\in [0,1]$ then, $\displaystyle (1-t)x+ty=((1-t)x_i+ty_i)$ satisfies $\displaystyle \sum_{i=1}^n(1-t)x_i+ty_i=(1-t)\sum_{i=1}^n x_i+t\sum_{i=1}^ny_i=(1-t)+t=1$ and $\displaystyle (1-t)x_i+ty_i\geq 0$ for all $\displaystyle i=1,\ldots,n$. That is, $\displaystyle (1-t)x+ty\in S_{n-1}$.