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Consider a pair of piecewise-linear cobordant $n$ dimensional polyhedra $P_1, P_2$ sitting in $\mathbb{R}^{n+2}$ (with some fixed orientation).

Let $O$ be an $n+1$ dimensional piecewise-linear cobordism such that $P_1, P_2$ mark the boundary of $O$.

Claim:

the number of $n+1$ dimensional facets of this cobordism plus 2, must be less than or equal to the number of $n+1$ dimensional facets of the convex hull of the two original polyhedra, with each of the original polyhedra treated as $n+1$ dimensional facets, if they lie on the boundary.

In loose terms I am attempting to argue that "a convex hull of a collection of polyhedra, when it contains said polyhedra on its boundary, is the simplest piecewise linear cobordism of the polyhedra".

I was considering an inductive argument: attempting to decompose a convex hull in $\mathbb{R}^{n+2}$ to "collection" of $n$ dimensional convex hulls, but I quickly run into problems there.

The other avenue, is to argue that a cobordism of minimal genus will have minimal number of facets (I don't know how to show this) and then try to make same argument involving Euler's Formula or perhaps something involving homology.


Noted by a friend, I am attempting to show the number of codimension-1 facets + 1 on a piecewise linear cobordism must be less than or equal to the codimension-1 facets of the convex hull of the two polyhedra, assuming they are each codimension-1 facets of their hull.

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