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Jan 14, 2023 at 6:01 comment added Ivan Izmestiev Usually this is stated in a broader context of quermassintegrals: the average volume of projections to $k$-dimensional linear subspaces is proportional to a certain mixed volume, which is a coefficient in the Steiner formula. I can mention two books: Santalo, Integral geometry and geometric probability, and Schneider, Convex bodies: the Brunn-Minkowski theory.
Jan 14, 2023 at 3:56 comment added Pluviophile Could you provide a reference on 1.?
Jan 12, 2023 at 17:46 history answered Ivan Izmestiev CC BY-SA 4.0