11 problems

Consider packings of unit balls in $\mathbb{R}^4$, and write $\Delta_4$ for the maximal packing density, $B_4$ for the volume of a unit ball, and $\delta_4=\Delta_4/B_4$ for the ma…

Wintgen's conjecture. Among tori in $\mathbb{R}^4$, the Willmore functional attains its minimum on the Clifford torus.

Let $P$ be a maximal kissing arrangement of points on the unit $3$-sphere with pairwise angular separation at least $\pi/3$, and let $(a,b)$ denote the parameters used in the paper…

Let $H$subset ${\bf R}^4$ be a closed half-space, and let $S$ be a unit sphere in $H$ tangent to the supporting hyperplane of $H$. The one-sided kissing number $B(4)$ is the maximu…

Four-dimensional logarithmic-correction conjecture. The accurate upper-bound scale for the length of $L$ is

Consider a fundamental domain of a lattice in ${\mathbb R}^4$ that contains a ball of radius $1$. The comparison object is a $24$–cell rescaled to have inradius $1$, and perimeter…

Consider an optimal 9-point configuration in the four-dimensional spherical-code problem. The notation $\{1,4,4^*\}$ denotes one point at a pole and eight vertices divided into two…

Let $P$ be an optimal packing of unit spheres in ${\mathbb R}^4$, with one sphere designated as the central sphere, and let $N$ be the number of neighboring spheres. The 24-cell co…

Let $P$ be a packing of unit spheres in ${\mathbb R}^4$, and consider its Voronoi decomposition. Let a Voronoi cell be the region associated with one sphere of the packing, and let…

Consider packings of unit balls in ${\Bbb R}^4$, and let the space-centred cubic lattice denote the specified four-dimensional lattice. Four-dimensional lattice-packing conjecture.…

Let $\Sigma$ be a surface in $\mathbb R^4$ with Willmore energy $W(\Sigma)$. Kusner's conjecture. If … then $\Sigma$ must be a sphere. This remains open in the context of the paper…