Simply put, my question is this: what is the smallest dimension, if any, where we can know for sure that a convex body exists whose translative packing constant is strictly larger than its lattice packing constant?
Some relevant facts I know:
In two dimensions, the translative packing constant is always equal to the lattice packing constant for convex bodies. For non-convex bodies, already in two dimensions there are counterexamples. Bezdek and Kuperberg give a good exposition of this.
In ten dimensions, the best packing of spheres seems to be non-lattice. In lower dimensions, the best lattice packing seems to be also the best packing. The best known lattice packing in dimensions above 8 is not actually known to be best (except in 24 dimensions). Therefore, even for spheres, there is no dimension where lattices are proved to be suboptimal. See this MO question for more on that.
Convex bodies that tile by translation can also tile by lattice translations. So the example cannot be a tiling body. This is famous result due to Venkov, Alexandrov, and McMullen, and Gruber's book on Convex and Discrete Geometry gives a nice treatment.
Except for tiling bodies, which are ruled out be the previous point, very few bodies have known translative packing constants. However, all we need to have an example is a lower bound for the translative packing constant and an upper bound for the lattice packing constant. There are known methods to compute the lattice packing constant for polytopes, and in 3D one such method has been implemented by Betke and Henk.
So, it seems that in dimension 10, there are convex bodies, namely spheres, that pack better by translation than by lattice translation. However, this is not rigorously known. If true, this example can probably be extended to higher dimensions by forming cylinders. However, it seems to me that if we allow nonspherical bodies, the dimension where translative packing starts to beat out lattice packing should be siginificantly lower. Is there an example? In dimensions low enough (e.g. 3D), candidates can be checked computationally and rigorously established if they check out.