Reference request on the relationship between inscribed polytopes and shadows of their facets

Question

Two chords of a circle of unit radius have equal lengths if their corresponding arcs have equal lengths.

Suppose a polytope is the convex hull of finitely many points on the unit $(n-1)$-sphere in $\mathbb R^n.$ Each of its $(n-1)$-dimensional facets casts a shadow on the sphere, the light-source being at the center. Suppose a proposition says two such facets have equal $(n-1)$-dimensional volumes if their shadows have equal volumes. I suspect that that is false in general, even if both facets have equally many vertices. (Unlike with chords of circles, their shapes can differ.)

My question is which books or exposititory articles give an introductory treatment of such questions as whether such a proposition as that is true.

Answer 1

This particular statement is false. When similar-sounding statements turn out to be true they can sometimes be proved by the methods of integral geometry / geometric probability, by comparing appropriately constructed measures.

Two sources are

1. "Integral Geometry and Geometric Probability" by Santalo,
2. "Introduction to Geometric Probability" by Klain and Rota.

Unfortunately, while it is not difficult to disprove the statement above, I have no nice disproof which demonstrates the use of integral geometry, so I hesitate to post this as an answer. But it seems better than no answer at all.