[ceh123]

On the topic of simplices! I did my PhD in dynamical systems and the space of invariant measures [0] is (in the compact setting) always a simplex and the extreme points are the ergodic measures. It's because of this that you can kind of assume your system is ergodic do work there and frequently be able to generalize to the non-ergodic case (through ergodic decomposition).

But the real thing I wanted to mention here was the Poulsen Simplex [1]. This is the unique Choquet simplex [2] for which the extreme points are dense. This means that it's like an uncountably infinite dimensional triangle where no matter where you are inside the triangle, you're arbitrarily close to a corner. It's my favorite shape and absolutely wild and impossible to conceptualize (even though I worked with it daily for years!)

[0] https://en.wikipedia.org/wiki/Invariant_measure

[1] https://eudml.org/doc/74350

[2] https://en.wikipedia.org/wiki/Choquet_theory