This paragraph should have had the highlighted words: << Back to K: As M approaches K smoothly (at least in the C^3 topology) the Gauss map is no longer well-defined at edges or vertices of M. But its image can be replaced by the set of outward unit normals to all planes P in space that (assuming K convex) that *intersect K and* keep K inside of one of the closed half-spaces defined by P. The area of the spherical image of all the unit normals must still be 2pi*X(K) = 4*pi.

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P.S. (This also works for nonconvex K but requires more words.)