# What is the average area of the shadow of a convex shape taken over all possible orientations?

If we take a sphere, its surface area is exactly 4 times the area of its shadow. How can it be generalised for any convex shape?

• Average measure of convex projections are proportional to their intrinsic (quermaß) volumes: see §7.4 “The mean projection formula” in Klain & Rota's delightful little book Introduction to Geometric Probability. – Gro-Tsen Apr 18 at 14:20
• In particular, yes, for a convex solid in $\mathbb{R}^3$, it is true that the surface area (which is essentially the $2$-dimensional intrinsic volume) is proportional to the area of the projection over all plane directions. – Gro-Tsen Apr 18 at 14:23
• Thank you but I guess I'm not ready to understand this §7.4, isn't there any other way to explain the at least the intuition? (I will read that book from the very beginning after all) – Betydlig Apr 18 at 14:42
• Then you had probably better ask on math.stackexchange.com instead (but remember to link to this post also, for completeness) — MathOverflow is for mathematicians to ask each other questions about their research. – Gro-Tsen Apr 18 at 14:47