It should be true that $n$ $2(n-1)$-simplices can be glued together, such that $k$ of them intersect in a common $(2n -k -1)$-cell and the resulting object is a "convex $2(n-1)$-disc". I am not sure how to make this more precise. For $n=1$ it's just a point, for $n=2$ we have two triangles glued together on a common side, and so on. Does the resulting object have a name?

It should be true that $n$ $2(n-1)$-simplices can be glued together, such that $k$ of them intersect in a common $(2n -k -1)$-cell and the resulting object is a "convex $2(n-1)$-disc" whose boundary $(2n-3)$-sphere contains $n^2$ $(2n-3)$-cells. I am not sure how to make this more precise. For $n=1$ it's just a point, for $n=2$ we have two triangles glued together on a common side, and so on. Does the resulting object have a name?

It should be true that $n$ $2(n-1)$-simplices can be glued together on common $(2n-3)$-cells, such that their boundary is a $(2n-3)$-sphere and they all intersect in a common $n$-cell. For $n=1$ it's just a point, for $n=2$ we have two triangles glued together on a common side, and so on. Does the resulting object have a name?

It should be true that $n$ $2(n-1)$-simplices can be glued together, such that $k$ of them intersect in a common $(2n -k -1)$-cell and the resulting object is a "convex $2(n-1)$-disc". I am not sure how to make this more precise. For $n=1$ it's just a point, for $n=2$ we have two triangles glued together on a common side, and so on. Does the resulting object have a name?