I read the article The Average Projected Area Theorem – Generalization to Higher Dimensions and learnt that Augustine Cauchy proposed in the 19th century that the average projected area of any convex solid is one fourth of its total surface area. The following table provides more general ratios:
More surprisingly, there is a nice closed form representation for the surface area $(k(d))$ in terms of dimension $d$. $$k(d) = \frac{1}{\sqrt{\pi}(d-1)M_d},$$ where $$M_d = \frac{\Gamma{\left(\frac{1}{2}(d-1)\right)}}{\Gamma{\left(\frac{1}{2}d\right)}}$$
Here are some of my motivations for posting this question.
1.The article does offer some explanation of how the formula is derived, but I do not totally understand it. I would appreciate a more detailed explanation.
2. I had a hard time visualizing projected area and surface area in higher dimension. Could people offer me some analogies or explanation to help me better understand concept like this?