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...

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

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 $\pi$...