Which convex pentagon gives least packing density? Among all convex pentagons, does the regular pentagon give least packing density?
Further question: For each $n > 6$, is the regular $n$-gon the minimum of packing density?
An analogous question can be asked on covering – for which values of $n= 5$ and above $6$, the regular n-gon gives the maximum covering density among all convex $n$-gons (ie. is the least economical for covering)?
Do the answers to these questions depend on central symmetry – whether $n$ is odd or even?
 A: The answer is (very likely) no for all $n>20$.
Let $R_n$ be the regular $n$-gon of circumradius $1$, and given a convex shape $C$, let $\delta(C)$ be its maximal packing density in the plane.
Observe that the packing density of $R_n$ is at least the packing density of the disk times $\text{Area}(R_n)/\pi$ (since we can just inscribe an $R_n$ into each disk).
Thus, we have:
$$\delta(R_n)\ge \frac{\sqrt{3}}6 n\sin(\pi/n)\cos(\pi/n) $$
On the other hand, the regular heptagon $R_7$ is conjectured to have maximal packing density $0.89269$. (The fact that even this result is conjectural should give a sense of how tricky it is likely to be to prove optimality for the regular pentagon - maximal packing densities are hard.)
However, at $n=21$, our bound gives a packing density of at least $0.893$, so at this and all higher $n$ we can do better by taking an $n$-gon collapsed to a regular heptagon.
Probably this result can be improved a bit by finding explicit packings for some $n<20$ which exceed $\delta(R_7)$, if desired (since the circle-inscribing method is never optimal). In fact, I would not be surprised if every $n>7$ could be packed better than $R_7$.
(If you don't want the reliance on a conjectural result, using the smoothed octagon in place of the regular heptagon will give you counterexamples past $n=37$ or so, though the specifics are a bit messier since one has to use a polygonal approximation to the smoothed octagon.)
A: A few years ago Tom Hales and Wöden Kusner determined the packing  density of the regular pentagon, see:
https://www.semanticscholar.org/paper/Packings-of-Regular-Pentagons-in-the-Plane-Hales-Kusner/7b91d1e27c67cc0b99540fa58fbee348dd16e102

Hales-Kusner Fig. 1.
The packing densities of other pentagons, with few exceptions, are unknown. It is known, however, that if a pentagon (convex or not) has a pair of parallel sides, then it tiles the plane with its congruent replicas. This seems to indicate that the regular pentagon, having its pairs of sides farthest from being parallel, is a likely candidate for the "worst packer" among all pentagons.
