Let C be an n-dimensional pointed convex polyhedral cone with a uniqe frame {a1.a2.......,ar}, where ais are extrem half-lines. Is there a formula for the number of r-faces in C?
Let me state the problem more concretely. Consider the case n=3, where the frame of a 3-dimensional pointed convex polyhedral cone C is denoted by $\left( {{a}_{1}} \right),\cdots \left( {{a}_{r}} \right)\ \left( r\ge n \right)$.，where (a) means a halfline spaned by a. It can be seen from the figure that $\left( {{a}_{i}}\ {{a}_{i+1}} \right),1\le i\le r,$ is 2-dimensional faces of C, if necessary, by renumbering, where i+1=1 if i=r. The number of n-1=2 dimensional faces is r. The number of n-2=1 dimensional faces is also r, since they lie on the common relative boundary of two adjacent 2 dimensional faces. The origin 0 is the only one n-3=0-dimensional face. The n=3 case can be seen from the figure and proved, but is there a formula for finding the number of faces in each dimension for n > 3?