What type of assumption would I have to make to get somewhere on this problem? Lovely reference, btw.

So $f > 0$ everywhere and assume that it has at least 1 local minima. In general it's nonconvex. I'd like the lower bound to depend on properties like $d$ and the number of critical points with the property $P$

Ok, so is this true in the case where $M$ is transverse to the faces of $C$?

Oops, didn't notice the edit.

Wait this isn't quiet what I'm looking for. I care about the embedding and the geometry of $M / \sim$, not just the topology. The circle $S^1 \subset \mathbb{R}^2$ also has a trivial tangent bundle, yet it's normals vary more than $90^{\circ}$. The normal spaces still have to vary as you travel along a path along the handle of the Clifford torus.

I don't believe that's true. Note that when I say "we can perturb $M$ so that it only intersects faces of $C$ of dimension $d−2$ or greater", I'm also saying that I assume $M$ intersects every face of $C$ transversally.

Wow, that is a weird object...

Yes, $M/\sim$ is compact.

I believe that M could be smoothed, which seems to be what the result you linked proves, but I don't see how this answer my question about the polyhedral complex.

Do you have a reference that might be useful?