Hadwiger’s illumination conjecture
Can every convex body in n dimensions have its entire boundary illuminated from at most 2ⁿ directions?
Discrete geometry
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Kissing number in five dimensions
How many equal five-dimensional balls can touch one central ball without overlapping each other?
Erdős–Ulam problem
Can points be dense throughout the plane while every distance between two of them is rational?
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?
Borsuk’s conjecture in dimension four
Can every bounded four-dimensional set be split into five pieces of strictly smaller diameter?
Conway’s thrackle conjecture
Can a planar thrackle have more edges than vertices?
Birkhoff billiard conjecture
Are ellipses the only smooth strictly convex planar billiards that are integrable near the boundary?
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?