I'm studying this example of nonshellable but constructible 3-ball on 10 vertices and 21 facets. Other than just staring at the pictures and hoping that they would eventually make sense, is there any other way to study them? Also if I have to explain this to someone else, what's the best way to do it?
Also, at the end of the construction the authors remark on some characteristics of the ball (e.g. none of the facets' intersection with the ball's boundary is a 2-ball, which shows that it's strongly nonshellable, or the fact that it can be split into 2 constructible 3-balls with a 2-ball as their intersection). Does this remark come about just by pure inspection and perhaps computer-assisted calculations (since the ball has a simple coordinate realization in $\mathbb{R}^3$)?
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Edit: I've just added the last part of the proof that the ball in question is strongly unshellable, where "strongly unshellable" is defined according the definitions and Proposition 2.4. (iii)-(iv) below.
Basically, the flow of argument requires me to check that none of the component tetrahedra has a 2-ball as its intersection with the "outer" surface of the 3-ball. Again I'm trying different ways to visualize this fact.
