It is well-known that the densest packing of circles in the plane is the close hexagonal packing, with a density of $\frac{\pi\sqrt{3}}6\approx0.9069$:
By applying an affine transformation, we obtain a packing of ellipses with the same density:
However, not every ellipse packing arises from such a transformation, as we can rotate the ellipses at different angles. So it does not follow that the ellipse must have the same packing density.
Are there any ellipses which have a higher packing density than the circle?
I realize this question may be very difficult to answer in the affirmative, given the subtlety and difficulty of most packing problems, but I am curious to hear of bounds on the problem (possibly as a function of the aspect ratio) and expert opinions on the likely answer, if nothing else.
It is worth noting that in the three-dimensional case, the packing density of an ellipsoid can be higher than that of the sphere, with some ellipsoids having packing densities of at least $0.7585\ldots$ compared to the sphere's $\frac{\pi}{\sqrt{18}}\approx0.7405$ (see Wolfram MathWorld, J. Wills).
Some further investigation reveals that ellipses do indeed have the same packing density as circles, though I hadn't seen it explicitly stated anywhere before so it seems worth leaving this question up for future readers.
In L. Fejes Tóth's 1949 paper Some packing and covering theorems, it is noted that a consequence of Theorem $1$ is that every centrally symmetric convex set has a packing density equal to its translational or lattice packing density, from which the desired result follows: we can assume that an optimal ellipse packing uses only translations, then apply an affine transformation to yield a circle packing of the same density, which could be improved were it any lower than $\pi\sqrt{3}/6$. (He mentions in footnote 8 that this result may have been independently given in an October 1949 lecture of K. Mahler.)
However, Tóth notes that there may still be non-lattice packings that attain the same density, so one can still ask whether there exist nontrivial deviations from the hexagonal packing with an optimal $\pi\sqrt{3}/6$ density. (I do not know whether this is the case, though I would be surprised to find it was true.)