Open-problem statements, with their sources attached.

If $n$ points in $\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are distance $1$ apart.

Does every convex polygon have a vertex with no other 4 vertices equidistant from it?

Erdős also conjectured that there is a $k$ for which every convex polygon has a vertex with no other $k$ vertices equidistant from it.

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 distances. Does $h(n)/n\to \infty$?

If $n$ distinct points in $\mathbb{R}^2$ form a convex polygon then some vertex has at least $\lfloor\frac{n}{2}\rfloor$ different distances to other vertices.

For sufficiently large n, is it the case that any set of n points with minimum distance $1$ that minimizes diameter must contain an equilateral triangle of side length 1?

Moser's Worm Problem* What is the minimal area (or greatest lower bound on the area) of a shape that can cover every unit-length curve?

Convex Moser's Worm Problem* What is the minimal area (or greatest lower bound on the area) of a convex shape that can cover every unit-length curve?

Let $x_1,\ldots,x_n\in \mathbb{R}^2$ and let $R(x_i)=\#\{ \lvert x_j-x_i\rvert : j\neq i\}$, where the points are ordered such that $R(x_1)\leq \cdots \leq R(x_n).$ Let $g(n)$ be the maximum number of distinct values the $R(x_i)$ can take. Is it true that $g(n) \geq (1-o(1))n$?

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$(1+c)\frac{n}{2}$ distinct distances determined between the $x_i$, for some constant $c>0$ and all $n$ sufficiently large?

In the spirit of related conjectures of Erdős and others, presumably some kind of assumption that the points are in general position (e.g. no three on a line and no four on a circle) was intended.

The no-k-in-line problem: For $N \geq k$ and $k > 2$, the AllowedSetSize is $(k - 1) N$, i. e. on an $N \times N$ subset, there is a set of $(k - 1) N$ points for which no $k$ lie on a line (and not such a set of bigger size).

Green's Open Problem 72 / No-three-in-line problem*: The no-k-in-line conjecture holds for $k = 3$.

Does the no-three-in-line problem hold when $N$ is big enough?

Given $n$ points in the unit disc, must there be a triangle of area at most $n^{-2+o(1)}$ determined by them?

Every convex set in $\mathbb R^3$ has $\mathrm{VC}_2$ dimension at most 1.

For every $n$ there exists some $d$ such that every convex set in $\mathbb R^{n + 1}$ has $\mathrm{VC}_n$ dimension at most $d$.

If $n \ge 2$, every convex set in $\mathbb R^{n + 1}$ has $\mathrm{VC}_n$ dimension at most 1.