Yesterday I bought a box of chocolates and remarked with a friend how they had just put enough in to get above a certain transparent window to the inside but everything else was empty space. I added that this was probably particularly bad because it doesn't seem like the spherical-shaped chocolates would pack very well (in comparison to, say, cubes).
I wondered if in fact the sphere is the worst "shape" to pack, but I'm not entirely sure what the most natural formalisation of this question is. The following is my first attempt, which I have no idea how to prove, but I am open to suggestion for better formalisations:
We will call a tile of $\mathbb{R}^n$ a convex compact subset and we will say two tiles are almost disjoint if their intersection is a subset of their boundaries. I can't necessarily remember what little measure theory I knew but probably we want some kind of normalisation condition and I think convex compact subsets are measurable, so let's say we also want tiles to have (Lebesgue) measure one.
Is it true that the tile with least packing density for any $n$ is the $n$-dimensional closed ball (of appropriate size to have volume 1)?
If we remove the convexity condition but say, we still want our tiles to be connected, my thoughts are that there is in fact no tile of least packing density as we can just take $n$-dimensional (closed) annuli which are thinner and thinner with increasing radii to retain having volume 1 and these have arbtrarily low packing density. Is this correct also?
EDIT: Based on a suggestion in the comments, I wish to clarify that "packing" here permits tiling by translation and rotation of the tiles.