Simplex

An $n$-simplex has $n+1$ affinely independent vertices.

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.

A face is the simplex spanned by a subset of the vertices.

The boundary subcomplex $\partial {\Delta }^n$ consists of all proper faces of the $n$-simplex. Its geometric realization of a simplicial complex is homeomorphic to the sphere $S^{n-1}$, while $|{\Delta }^n|$ is homeomorphic to the closed disk $D^n$.

A regular simplex has pairwise equidistant vertices. Conversely, every finite equidistant subset of Euclidean space is the vertex set of a regular simplex.