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Consider a dense sphere packing in $\mathbb{R}^n$, i.e. an arrangement of mutually disjoint solid open spheres, all of the same radius.

In dimensions $2, 3, 8,$ and $24$, it is known that lattice packings (packings where center of the spheres form a discrete subgroup of $\mathbb{R}^n$) are optimal.

It is widely believed that, in high enough dimensions, the best packings will be non-lattice packings. Is anything known about the best non-lattice packing? Do the methods of Viazovska et al. and Hales give values for the density of the best non-lattice packing? Is anything known (or conjectured) about the the ratio of the density of the best non-lattice packing to the density of the best lattice packing?

Consider a dense sphere packing in $\mathbb{R}^n$, i.e. an arrangement of mutually disjoint solid open spheres, all of the same radius.

In dimensions $2, 3, 8,$ and $24$, it is known that lattice packings (packings where center of the spheres form a discrete subgroup of $\mathbb{R}^n$) are optimal.

It is widely believed that, in high enough dimensions, the best packings will be non-lattice packings. Is anything known about the best non-lattice packings? Do the methods of Viazovska et al. and Hales give values for the density of the best non-lattice packing? Is anything known (or conjectured) about the the ratio of the density of the best non-lattice packing to the density of the best lattice packing?

Consider a dense sphere packing in $\mathbb{R}^n$, i.e. an arrangement of mutually disjoint solid open spheres, all of the same radius.

In dimensions $2, 3, 8,$ and $24$, it is known that lattice packings (packings where center of the spheres form a discrete subgroup of $\mathbb{R}^n$) are optimal.

It is widely believed that, in high dimensions, the best packings will be non-lattice packings. Is anything known about the best non-lattice packing? Do the methods of Viazovska et al. and Hales give values for the density of the best non-lattice packing? Is anything known (or conjectured) about the the ratio of the density of the best non-lattice packing to the density of the best lattice packing?

Consider a dense sphere packing in $\mathbb{R}^n$, i.e. an arrangement of mutually disjoint solid open spheres, all of the same radius.

In dimensions $2, 3, 8,$ and $24$, it is known that lattice packings (packings where center of the spheres form a discrete subgroup of $\mathbb{R}^n$) are optimal.

It is widely believed that, in high enough dimensions, the best packings will be non-lattice packings. Is anything known about the best non-lattice packing? Do the methods of Viazovska et al. and Hales give values for the density of the best non-lattice packing? Is anything known (or conjectured) about the the ratio of the density of the best non-lattice packing to the density of the best lattice packing?

Consider a dense sphere packing in $\mathbb{R}^n$, i.e. an arrangement of mutually disjoint solid open spheres, all of the same radius.

In dimensions $2, 3, 8,$ and $24$, it is known that lattice packings (packings where center of the spheres form a discrete subgroup of $\mathbb{R}^n$) are optimal.

It is widely believed that, in high dimensions, the best packings will be non-lattice packings. Is anything known about the best non-lattice packing? Do the methods of Viazovska et al. and Hales give values for the density of the best non-lattice packing? Is anything known (or conjectured) about the the ratio of the density of the best non-lattice packing to the density of the best lattice packing?

EDIT: I suppose we don't necessarily know that the supremum over all possible non-lattice packings is attained, for example in $1$ dimension I believe non-lattice packings can get arbitrarily close the density $1$ of a lattice packing, but there doesn't seem to be a non-lattice packing with density $1$.

Consider a dense sphere packing in $\mathbb{R}^n$, i.e. an arrangement of mutually disjoint solid open spheres, all of the same radius.

In dimensions $2, 3, 8,$ and $24$, it is known that lattice packings (packings where center of the spheres form a discrete subgroup of $\mathbb{R}^n$) are optimal.

It is widely believed that, in high dimensions, the best packings will be non-lattice packings. Is anything known about the best non-lattice packing? Do the methods of Viazovska et al. and Hales give values for the density of the best non-lattice packing? Is anything known (or conjectured) about the the ratio of the density of the best non-lattice packing to the density of the best lattice packing?