This paper deals with your question: https://cs.uwaterloo.ca/~alubiw/papers/cauchy.pdf On Tue, Feb 2, 2021 at 14:54 Dan Asimov <asimov@msri.org> wrote:
Suppose P is a compact convex polyhedron in 3-space, topologically equivalent to the sphere, with finitely many convex planar faces each bounded by a convex polygon with finitely many straight sides.
Then every point x of P that is not a vertex has a neighborhood that is isometric to a neighborhood in the plane. And each vertex is locally isometric to a neighborhood of the vertex of a cone, with total angle around the vertex* equal to some theta in the open interval (0, pi).
For the jth vertex, call this number theta_j.
If this cone is approximated by a smooth surface then its unit normals translated to the origin will cover an area of the unit sphere equal to 2pi - theta. So if we assign the number
2pi - theta_j
to the jth vertex, the Gauss-Bonnet theorem tells us that
Sum (2pi - theta_j) = 4pi
or equivalently
Sum theta_j = 2pi(V-2)
if V is the number of vertices.**
Now suppose we have a finite number of regions P_j in the plane, each bounded by a polygon with straight sides, and that we are given a pairing of all the edges with edges of equal length so that, if the resulting abstract surface has V vertices, E edges, and F faces, the following two conditions are satisfied:
1) V - E + F = 2
(The Euler characteristic of the resulting surface is 2, from which we know that it must be a sphere.)
and
2) The total angle about vertex j is theta_j in the interval (0, pi), such that
Sum theta_j = 2pi(V-2).
Question: --------- Does there then necessarily exist a convex polyhedron in 3-space that is isometric to the resulting abstract surface?
—Dan
————— * The total angle of a cone can be seen if you slit the cone from the base to the vertex and flatten it out, it will fill a certain fraction f of a disk. Then its total angle is 2pi*f.
** E.g., for a cube V = 8 and each theta_j = 3pi/2, and indeed 8(3pi/2) = 2pi(8-2). _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun