Firstly, let’s examine more details about the “In” triangles.
In(Cone A, Roof* A) is the set of all the slant triangles under Roof* A.
And we denote the projection on the hyperplane by In(A). That is, In(A) is the set of all the flat triangles whose corresponding slant triangle belongs to In(Cone A, Roof* A), as shown on the left.
Shown in the middle figure is the relative position of slant triangles and the conjugate roof Roof* A. The blue plane denotes a surface of Roof* A and arrows indicate the direction of “above” and “under” the surface.
And, as shown on the right, In(Cone A, Roof* A) consists of the slant triangles whose vertices are on or under the surface of Roof* A.
For example, the left figure (of the figure ) shows the bottom triangle (colored red), whose top vertex is on the surface and the others are under the surface.
The middle figure shows the second triangle from the bottom, whose second vertex is on the surface and the others are under the surface.
The right figure shows the third triangle from the bottom, whose second vertex is on the surface and the others are under the surface.