Probability simplex (${\Delta }_n=\{x\in {\mathbb{R}}^n:x\ge 0,1^{T}x=1\}$)

The probability simplex consists of vectors of nonnegative weights summing to one. It is the convex hull of the coordinate vectors. A linear function $r^{T}x$ has maximum ${\max }_i{r}_i$ on this simplex; equality holds exactly when the weights are supported on maximizing indices.