Sphere packing in five dimensions
How densely can identical nonoverlapping spheres fill five-dimensional Euclidean space?
Geometric extremes
Packing, covering, distances and other limits set by geometry.
Subjects: Discrete geometry · Convex geometry · Algebra and geometry · Analysis and dynamics
Problems in this collection
11 problemsKissing number in five dimensions
How many equal five-dimensional balls can touch one central ball without overlapping each other?
Hadwiger’s illumination conjecture
Can every convex body in n dimensions have its entire boundary illuminated from at most 2ⁿ directions?
Symmetric Mahler conjecture
Do cubes and their dual shapes minimize a symmetric body’s volume multiplied by its polar’s volume?
Borsuk’s conjecture in dimension four
Can every bounded four-dimensional set be split into five pieces of strictly smaller diameter?
Moser’s convex worm problem
What is the smallest-area convex region that can contain every planar curve of length one after repositioning it?
Minimum opaque barrier for a square
How little total barrier length can intercept every straight line that passes through a unit square?
Cartan–Hadamard conjecture
Must a region in a nonpositively curved space have at least the boundary area of a Euclidean ball with the same volume?
Gersho’s conjecture in three dimensions
For many representative points in a uniform three-dimensional region, is a body-centered cubic arrangement asymptotically best at minimizing squared distance to the nearest representative?
Kelvin problem
What periodic partition of three-dimensional space into equal-volume cells uses the least shared surface area?
Erdős distinct distances problem
Must every set of n distinct points in the plane determine at least a constant times n/√log n different distances?