Unsolved Problems

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...

Problem Classify the point sets with no empty pentagon....

Conjecture For any integer $n \geq 1$, it is impossible to cover a square of side greater than $n$ with $n^2+1$ unit squares....

Problem Enumerate all convex uniform 5-polytopes....

Throughout this post, by projective plane we mean the set of all lines through the origin in $\mathbb{R}^3$. Definition Say that a subset $S$ of the ...

Conjecture For every set $P$ of $n$ points in the plane, not all collinear, there is a point in $P$ contained in at least $\frac{n}{2}-c$ lines determ...

Question What is the least integer $f(n)$ such that every set of at least $f(n)$ points in the plane contains $n$ collinear points or a subset of $n$ ...

Conjecture For each $\ell\geq3$ there is an integer $f(\ell)$ such that every set of at least $f(\ell)$ points in the plane contains $\ell$ collinear ...

Conjecture Associahedra have unbounded chromatic number....

To divide a given 2D convex region C into a specified number n of convex pieces all of equal area (perimeters could be different) such that the total ...

Conjecture Every convex polyhedron has a (nonoverlapping) edge unfolding....

Conjecture Let $k$ be a field of characteristic zero. A collection $f_1,\ldots,f_n$ of polynomials in variables $x_1,\ldots,x_n$ defines an automorphi...

Conjecture Let $X$ be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subva...

The fatness of a 4-polytope $P$ is defined to be $(f_1 + f_2)/(f_0 + f_3)$ where $f_i$ is the number of faces of $P$ of dimension $i$. Question Does ...

Conjecture The order of the largest total curvature of the primal central path over all polytopes defined by $n$ inequalities in dimension $d$ is $n$....

Conjecture For every positive integer $k$, there exists an integer $d$ so that every polytope of dimension $\ge d$ has a $k$-dimensional face which is...

The extension complexity of a polytope $P$ is the minimum number $q$ for which there exists a polytope $Q$ with $q$ facets and an affine mapping $\pi$...

Conjecture Every convex polytope has a non-overlapping edge unfolding....