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Hi, this is another question from the article "The Mathematics of Doodling" in the Mathematical Monthly of February 2011. (See http://math.stanford.edu/~vakil/files/monthly116-129-vakil.pdf)

On pp 126, the article mentions a remarkable fact:

Theorem 3. The average length of the shadow of a convex region of the plane, multiplied by $\pi$, is the perimeter.

Followed by:

Theorem 4. Consider the average area of the shadow of a convex region of three-space, and multiply by 4. The result is the surface area.

Are these well-known facts? I haven't heard of these facts before! Any ideas on how one could prove these two results?

Thank you in advance.

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1  
Yes these facts are well known, google "integral geometry"... –  Anton Petrunin Sep 17 '11 at 3:25
    
See also, "The Average Projected Area Theorem -- Generalization to Higher Dimensions," by Zachary Slepian, arxiv.org/abs/1109.0595 , posted just a few weeks ago. –  Joseph O'Rourke Sep 17 '11 at 11:29
    
In particular, there's Klain and Rota's book, "Geometric Probability" –  Deane Yang Sep 17 '11 at 12:51
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I would add that Slepian seems unaware that the higher dimensional formula, known as the Cauchy surface area formula, has been known for a rather long time. –  Deane Yang Sep 17 '11 at 12:55
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But Slepian appears to be an undergraduate physics major at Princeton, so I guess he should be forgiven for his oversight. Anyway, proofs of the Cauchy surface area formula (in all dimensions) can indeed be found in the books of Klain and Rota (at least for polytopes), Schneider, and Santalo. I don't know who first extended Cauchy's proof from dimensions 2 and 3 to higher ones. –  Deane Yang Sep 17 '11 at 15:38

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