I'm interested in the following question, which seems to be assumed all over the place (at least for 3 dimensions) in convex geometry, and which I cannot find a proof of.

Suppose we have a corner of a convex polytope in $\mathbb{R}^d$. How do we show that we can 'flatten' the surrounding facets into $d-1$ dimensions without overlap?

What do I mean by that? Well, for the three-dimensional case, it means that if you sum the angles around a given vertex of a convex polyhedron, you get a sum of less than $2 \pi$. Here's another way to say that:

(0) Given a collection of 2-dim cones $C_i$ with angles $a_i$, if the sum of the $a_i$ is greater than $2 \pi$, then $C_i$ cannot be facets of a 3-dim cone.

(I'm defining a cone to be the convex hull of a collection of rays; so, cones are assumed to be convex.)

And in general dimensions:

(1) Given a collection of $(d-1)$-dim cones $C_i$ with total $(d-1)$-angle measures $a_i$, if the sum of the $a_i$ is greater than the total angle surrounding a point in $\mathbb{R}^{(d-1)}$, then the $C_i$ cannot be facets of a d-dim cone.

This fact can be restated in a lot of other ways. Maybe one of these is easier to prove?

(2) If a collection of $(d-1)$-dim cones $C_i$ with total $(d-1)$-angle sum greater than the total angle surrounding a point in $\mathbb{R}^{(d-1)}$ is configured in $\mathbb{R}^d$ with all cone points set at the origin and every $(d-2)$-face identified with a $(d-2)$-face of some other cone (i.e., the cones are glued to make a simplicial complex), and if this configuration lies on one side of a hyperplane, then the configuration is not convex.

(3) The facets of any d-cone can be isometrically mapped (unfolded!) into a $(d-1)$-hyperplane, retaining the coincidence of the cone point and without overlap.

(4) The convex spherical polygon of largest perimeter is a great circle, i.e. any spherical polygon with perimeter larger than $2 \pi$ is not convex.

It seems to me like there should be a straightforward, convex-geometry proof of this fact, but I can't find it. If you know another way to prove it (say, using ideas from curvature?) I'd be very interested in that too!