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The average projected area theorem states that for a convex body in 3D the average projected area $\langle \sigma\rangle$ for a random orientation is 1/4 of the surface area $A$ (despite the PDF often having complex shapes). It has nice generalisations to higher dimensions.

For non-convex contiguous bodies the ratio $\langle \sigma\rangle/A$ can clearly be made arbitrarily small, even if we prescribe a fixed volume $V$ of the body (just turn it into a very long and thin line). But is there some counterpart to the theorem if the non-convex body is thickened by adding all points within radius $r$?

In the line case there will now be a minimum $\sigma=\pi r^2$ and side views running up to $\sigma=2rL$; indeed, in this case since the new body is convex the theorem gives $\langle \sigma\rangle = \pi r ( L/2 + r)$. If we instead have $N$ "antennas" from a central point in random directions of length $L$ the projected area will be $\langle \sigma\rangle \approx N(\pi L/4)(2r)$ (the $\pi/4$ factor is from the random orientation of the segments; I am ignoring the end-spheres and central overlap) while the surface area will be $A\approx 2\pi r LN$, giving $\langle \sigma\rangle / A \approx 1/4$.

But clearly one can get smaller ratios if there are internal spaces or folds of the surface with internal diameter $>2r$. For sufficiently large $r$ we get a convex body where $\langle \sigma\rangle / A = 1/4$, but as $r$ decreases the ratio can decrease. Is there some way of bounding this decrease?

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    $\begingroup$ The ratio can be arbitrarily small: take the solid to be a series of nested spherical shells separated by the appropriate distance (with holes or connecting spokes added as desired to address any topological concerns). A natural way to ask about this decrease might be to fix the convex body to be of unit diameter, and look at bounds for each value of $r$? Even for $r>1$ the ratio won't necessarily be $1/4$, though; consider a starting set of two points, or for a connected example a sort of "barbell" shape (two discs with a rod between their centers). These are non-convex for all $r$. $\endgroup$ Dec 20, 2020 at 5:30

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