40 problems

Let $K\subset\mathbb R^{2}$ be a centrally symmetric convex body, and let $\delta_{L}(K)$ denote the density of its densest lattice packing. The Reinhardt conjecture. Up to an affi…

Let $\mathsf{E}_8$ denote the root lattice in dimension $8$. A globally optimal solution to the lattice packing-covering problem is one attaining the best packing-covering value am…

Let $\mathsf{E}_8$ denote the root lattice in dimension $8$. A lattice is a locally optimal solution to the lattice packing-covering problem if no sufficiently small deformation im…

A perfect form is a positive definite quadratic form whose associated Delone subdivision is determined by its minimal vectors; Voronoi's first perfect form is the first form in Vor…

Let $\mathcal{S}$ be the densest lattice packing of spheres in $\mathbb{R}^4$, composed of fcc layers. Let $H$ be a hyperplane in an fcc direction, and let $\widehat{\mathcal{S}}$…

For each dimension $n$, let $\Delta_n$ be the highest possible density of a packing of equal nonoverlapping spheres in $\mathbb R^n$, and let $\Delta_n^{(L)}$ be the highest densit…

Reciprocal counting-function conjecture. For every $x$, one has

Weighted counting-function conjecture. For $x\geq 8$, one has

Schmutz Schaller's counting-function conjecture. For every $x$, one has

Cohn–Rajagopal's conjecture. Every such valid four-colored configuration has center density at most $1$.

A lattice packing in Euclidean space is a sphere packing whose centers form a lattice, and a lattice packing is suboptimal when its density is strictly less than the optimal densit…

Perturbation conjecture. We moreover conjecture that

Let $n\geq 2$ and let $K\subset\mathbb R^n$ be an origin-symmetric convex body satisfying … where $c>0$ is a universal constant. Convex-body extension conjecture. There exists a la…

Cross-polytope kissing number conjecture. In $\mathbb{E}^4$,

Cross-polytope lattice density conjecture. In $\mathbb{E}^4$,

Let $B^n$ denote the unit ball in Euclidean space $\mathbb{E}^n$, and let $\kappa^*_{\alpha}(B^n)$ be the generalized lattice kissing number at parameter $\alpha$. Conjecture 8.1.…

A close-packing in dimension 4 is a packing by the four-dimensional Lee spheres described in the surrounding construction, with the case under discussion arising from the classical…

Suppose that $D^2=2\ell^2$ with $\ell\in\mathbb N$ and $\ell=0\mod 3$. There are at least two $\mathbb Z^3$-symmetry classes of $D\cdot\mathbb A_3$-sub-lattices, and each such sub-…

Let $D^2=2\ell^2$ with $\ell\in\mathbb N$, $\ell\neq 0\mod 3$, and $\ell\neq 2^n$. In this case the perfect configurations consist of $D$-FCC sub-lattices and their $\mathbb Z^3$-s…

Let $D^2=2\ell^2$ with $\ell$ divisible by $3$, and let a given $D\cdot\mathbb A_3$-sub-lattice admit a continuum family of layered dense-packings. A perturbation analogous to the…

For a convex body $K\subset\mathbb R^{2}$, let $d_{2,1}(K)$ be the non-separable lattice quantity defined in the source. The Makai extremal conjecture. … with equality possible onl…

Let $K\subset\mathbb R^{2}$ be a convex body, and let $d_{2,1}(K)$ denote the quantity defined in the source as the minimum density of a non-separable lattice of translates of $K$.…

A lattice packing in $\mathbb{R}^4$ has density bounded above by the lattice three-point bound, obtained from a semidefinite-programming formulation of the three-point bound. Sharp…

Let $\code{C}^\ast$ be a formally self-dual code of length $n$. Let $f_{\code{C}}(t)$ be the polynomial associated with a formally self-dual code $\code{C}$, and let…

Let $\Gamma$ be a formally unimodular packing, and let $\Xi_{\Gamma}(\tau)$ denote its secrecy function and $\xi_{\Gamma}$ its maximum value. Formally unimodular packing secrecy fu…