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?...

Let $h(n)$ be such that any $n$ points in $\mathbb{R}^2$, with no three on a line and no four on a circle, determine at least $h(n)$ distinct distance...

Let $A$ be a set of $n$ points in $\mathbb{R}^2$ such that all pairwise distances are at least $1$ and if two distinct distances differ then they diff...

Let $h(n)$ count the number of incongruent sets of $n$ points in $\mathbb{R}^2$ which minimise the diameter subject to the constraint that $d(x,y)\geq...

Let $\alpha(n)$ be such that every set of $n$ points in the unit disk contains three points which determine a triangle of area at most $\alpha(n)$. Es...

Let $x_1,\ldots,x_n\in \mathbb{R}^2$ and let $R(x_i)=\#\{ \lvert x_j-x_i\rvert : j eq i\}$, where the points are ordered such that $ R(x_1)\leq \cdots...

Let $x_1,\ldots,x_n\in \mathbb{R}^2$ and let $R(x_i)=\#\{ \lvert x_j-x_i\rvert : j eq i\}$, where the points are ordered such that $ R(x_1)\leq \cdots...

Let $x_1,\ldots,x_n\in \mathbb{R}^2$ be such that no circle whose centre is one of the $x_i$ contains three other points. Are there at least $ (1+c)\f...

Are there, for all large $n$, some points $x_1,\ldots,x_n,y_1,\ldots,y_n\in \mathbb{R}^2$ such that the number of distinct distances $d(x_i,y_j)$ is $...

Consider the triangular lattice with minimal distance between two points $1$. Denote by $f(t)$ the number of distances from any points $\leq t$. For e...

Let $F_k(n)$ be minimal such that for any $n$ points in $\mathbb{R}^2$ there exist at most $F_k(n)$ many distinct lines passing through at least $k$ o...

Let $h(n)$ be maximal such that in any $n$ points in $\mathbb{R}^2$ (with no three on a line and no four on a circle) there are at least $h(n)$ many c...

For $x_1,\ldots,x_n\in [-1,1]$ let $ l_k(x)=\frac{\prod_{i eq k}(x-x_i)}{\prod_{i eq k}(x_k-x_i)}, $ which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$...

Conjecture Let $p_1,p_2,p_3,\ldots$ be a sequence of points in ${\mathbb R}^d$ with the property that for every $i \ge d+2$, the points $p_{i-1}, p_{i...

Let $S$ be a set of points in the plane. Two points $v$ and $w$ in $S$ are visible with respect to $S$ if the line segment between $v$ and $w$ contain...

Conjecture The average diameter of a bounded cell of a simple arrangement defined by $n$ hyperplanes in dimension $d$ is not greater than $d$....

Basic Question: Given any positive integer n, can any convex polygon be partitioned into n convex pieces so that all pieces have the same area and sam...

Problem Does there exist a dense set $S \subseteq {\mathbb R}^2$ so that all pairwise distances between points in $S$ are rational?...

Question What is the minimum cardinality of a decomposition of the $n$-cube into $n$-simplices?...

Conjecture If a finite set of unit balls in $\mathbb{R}^n$ is rearranged so that the distance between each pair of centers does not decrease, then the...

Conjecture Every set of $2^{n-2} + 1$ points in the plane in general position contains a subset of $n$ points which form a convex $n$-gon....

If $X \subseteq {\mathbb R}^2$ is a finite set of points which is 2-colored, an empty triangle is a set $T \subseteq X$ with $|T|=3$ so that the conve...

Question Is is possible to pack $n^n$ rectangular $n$-dimensional boxes each of which has side lengths $a_1,a_2,\ldots,a_n$ inside an $n$-dimensional ...

Question What is the minimum number of colours such that every complete geometric graph on $n$ vertices has an edge colouring such that: \item[Varian...

Conjecture Every complete geometric graph with an even number of vertices has a partition of its edge set into plane (i.e. non-crossing) spanning tree...