I am currently working on research involving packing problems and am finding myself needing the tools from Combinatorial Geometry (in particular, I've been reading Pach and Agarwal's book on the subject) and I am in the dark on what I think should be a very simple point. I apologize if the question is too elementary for MO.
For reasons I won't get into, I am needing to give a bound for the number of spherical 2-simplexes which can occur among n-points in S2, as this will tell me how many exposed faces of a simplicial 3-complex there are among a certain subset of n-points in E3. I am at a point in a Lemma where I am needing to generalize the following claim about graphs in the plane, to simplicial 2-complexes (graphs) in S2.
I cite from Pach and Agarwal's Combinatorial Geometry: "The internal angle of a simple closed polygon C which bounds a graph G at a vertex of degree d is at least (d−1)π3."
Does anyone know a proof of this fact, or have a simple explanation of it so that I can have a hope of generalizing it to "the internal angle of a simple closed spherical polygon which bounds a simplicial 2-complex at a vertex of degree d is at least [something] in S2".
Thank you, I appreciate any responses.
EDIT: Due to Joseph O'Rourke's comment I will quote a larger passage from the book so that there is more context.
The properties of a graph G are being discussed and the following is mentioned:
The outer face of G is bounded by a simple closed polygon C. Let b and bd denote the total number of vertices of this polygon and the number of those vertices that have degree d in G, respsectively. Clearly, b=b2+b3+b4+b5. The internal angle of C at a vertex of degree d is at least (d−1)π3, and the sum of these angles is (b−2)π. Hence, b2+2b3+3b4+4b5≤3b−6.
My interpretation of this is that we look at the boundary of our graph G, and note that it is a closed polygon. The angles of this polygon at a particular vertex (namely, one of degree d in the graph G) is the "the internal angle of C at a vertex of degree d"
EDIT 2: I apologize, there seems to be more background that I need to mention in order for this to make sense. As mentioned in my comment directed towards Will Jagy, the following condition is to hold for our graph G in the plane. Consider a set P of n points with minimum distance 1, and connect two elements of P by a segment if and only if their distance is exactly 1. Thus, we obtain a graph G embedded in the plane. For my case, I want to consider a set P⊆S2 of n points with minimum distance π3, and connect two elements of P by a great arc if and only if their distance is exactly π3.
Hopefully that was the last edit, but I will reiterate exactly what my question is.
Where does the condition come from in E2 that, "the internal angle of C at a vertex of degree d is at least (d−1)π3". Can a similar condition be generalized to my case on S2?