There's also a 2d version: the average of all the widths ("diameters") of a convex shape in the plane is 1/π times the perimeter. The circle is a trivial case because all widths are diameters: C=πd.
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The unit cube has surface area 6. So the average area of a unit cube's shadow is 3/2.
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Can I assume this only holds when the "light source" is a point at infinity, i.e. parallel rays?
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Yes, exactly.
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I once used this formula to approximate the size of blob shadow for objects w.r.t. the sun in an old 3d graphics engine project. It was delightful to see it mentioned again.
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You just earned yourself a follower!
Hvala. Twitter će to iskoristiti za poboljšanje vaše vremenske crte. PoništiPoništi
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What sorcery is this?!? (admittedly, i missed "convex" on first reading and immediately thought of some counterexamples. With "convex" firmly in place mentally, now i'm just surprised and amazed that there's a universal limit.)
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I learned about “quermassintegrals” and Cauchy-Crofton formulas in excellent books on Integral Geometry written by people like Fenchel and Santalo, many years ago.
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The dark shadow cast by such an object is known as a Cauchy schwarz.
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Hvala. Twitter će to iskoristiti za poboljšanje vaše vremenske crte. PoništiPoništi
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