Before we can apply more advanced fixed point theorems to the Evasiveness Conjecture, we need a little background on simplicial complexes, and everything starts with simplices.

**Simplices**

It’s most intuitive to start with the geometric viewpoint, in which case an *-simplex* is defined to be the convex hull of affinely independent points in . These points are called the *vertices* of the simplex. Here are examples for

A *simplicial complex*** **is then a collection of simplices glued together along lower-dimensional simplices. More formally, if is a (geometric) simplex, then a *face of * is a subset formed by taking the convex hull of a subset of the vertices of .

Finally, a *(geometric) simplicial complex* is a collection of simplices such that

- If and is a face of , then , and
- If and , then is a face of both and .

Property (1) gives us downward closure, and property (2) specifies how simplices can be glued together (only along faces). For instance, the first picture depicts a simplicial complex. The second does not.