Unsolved Problems

Does a convex body in $\mathbb{R}^n$ with one interior lattice point at its center of mass have volume at most $(n+1)^n/n!$?...

Can every bounded set in $\mathbb{R}^n$ be partitioned into $n+1$ sets of smaller diameter?...

What is the maximum number of non-overlapping unit spheres that can touch a central unit sphere in $n$ dimensions?...

Is the sphere the worst-packing convex solid?...

What is the densest packing of unit spheres in dimensions other than 1, 2, 3, 8, and 24?...

Is there a dense set of points in the plane with all pairwise distances rational?...

Must a Kakeya set in $\mathbb{R}^n$ have Hausdorff and Minkowski dimension $n$?...

Can every convex body in $\mathbb{R}^n$ be illuminated by $2^n$ light sources?...

Does Yang-Mills theory exist mathematically and exhibit a mass gap in 4D?...

What is the kissing number (maximum number of non-overlapping unit spheres that can touch a central unit sphere) in dimensions other than 1, 2, 3, 4, ...

For n > 14 points (except n=24), what is the maximum minimum distance between points on a unit sphere?...

Does every convex, closed, twice-differentiable surface in 3D Euclidean space have at least two umbilical points?...

Does the isoperimetric inequality extend to Cartan-Hadamard manifolds (complete simply-connected manifolds of nonpositive curvature)?...

Does the Euler characteristic of a compact affine manifold vanish?...

What are the relationships between curvature and Euler characteristic for higher-dimensional Riemannian manifolds?...

Is the first eigenvalue of the Laplace-Beltrami operator on an embedded minimal hypersurface of $S^{n+1}$ equal to $n$?...

Can every $n$-dimensional convex body be covered by at most $2^n$ smaller positively homothetic copies?...

What is the minimum number $g(n)$ of points in general position in the plane guaranteeing a convex $n$-gon?...

What configuration of $n$ points in the unit square maximizes the area of the smallest triangle they determine?...

Does every centrally symmetric $d$-dimensional polytope have at least $3^d$ faces?...

What is the maximum number of 3-point lines attainable by a configuration of $n$ points in the plane?...

How many pairs of points at unit distance can be determined by $n$ points in the Euclidean plane?...

What is the shortest path that guarantees reaching the boundary of a given shape, starting from an unknown point with unknown orientation?...

Can three unknotted space curves (not all circles) be arranged as Borromean rings?...

Do Danzer sets of bounded density or bounded separation exist?...

What is the optimal sphere packing density in dimensions >3?...