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?