What is the average projected area of a thickened non-convex body?

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?

• 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 $r$. Dec 20, 2020 at 5:30