#I guess you could also use cot(πp) or the inverse of any other sigmoid function

[pantheraleo04]

Based on this, a sharp concave corner would be less spiky than one that's nearly flat. To me, a corner with an internal angle of like 345° still feels spiky in a way. At the very least it can't be called smooth right?

What if you measured spikyness with some function like s=(p-0.5)/(p²-p) where p is the probability you described. Then it can nicely describe corners that spike outward or inward. And a point that's completely smooth would have 0 spikyness.

Here's a measure of spikiness that can be compared between dimensions.

Take some corner you'd like to find the spikiness of.

If you stand at the corner and go in a random direction (picked uniformly from the unit sphere), what is the chance you go inside the corner?

That is the pure form of the spikiness measurement. You can take its reciprocal to make the unit more usable. I'll call this unit the Spikiness of the corner.

For example, a regular hexagon's corner has a Spikiness of 3, a cube's corner has a Spikiness of 8, a tetrahedron's corner has a Spikiness of I believe around 24, and a 5D cube's corner has a Spikiness of 32.