12 problems

If $C,D\subseteq \mathbb{R}^2$ then the distance between $C$ and $D$ is defined by $\delta(C,D)=\inf_{\substack{c\in C\\ d\in D}}\| c-d\|.$ Let $h(n)$ be the maximal number of un…

If $A$ is a finite set of $n$ distinct points in $\mathbb{R}^2$ in convex position, does $A$ determine at least $\left\lfloor n/2\right\rfloor$ distinct Euclidean distances?

Let $P$ be a finite set of points in $\mathbb{R}^2$ whose points are in convex position. Let $\{u_1,\ldots,u_t\}$ be the set of distances determined by pairs of points of $P$, and…

Let $n\ge3$ distinct points $p_0,\ldots,p_{n-1}\in\mathbb R^2$ form a convex polygon. Must there exist a vertex $p_i$ having at least $\lfloor n/2\rfloor$ distinct distances from t…

Let $f(n)$ be maximal such that any $n$ points in $\mathbb{R}^2$, with no three on a line, determine at least $f(n)$ different convex subsets. Estimate $f(n)$ - in particular, does…

Let $x_1,\ldots,x_n\in \mathbb{R}^3$ be the vertices of a convex polyhedron. Are there at least $(1-o(1))\frac{n}{2}$ many distinct distances between the $x_i$?

For $d\geq3$, let $ES_d(n)$ be the least integer such that every set of $ES_d(n)$ points in general position in $\mathbb{R}^d$ contains $n$ points in convex position. Is…

For each $k$, does there exist $g(k)$ such that every set of $g(k)$ planar points in general position contains $k$ points in convex position whose convex hull contains no other poi…

Let $f(n)$ be the least integer $N$ such that every set of $N$ points in $\mathbb{R}^2$, with no three points collinear, contains $n$ points that are the vertices of a convex $n$-g…

For a finite set $A\subset\mathbb{R}^2$, a point $p$ has at least $n$ equidistant points in $A$ if there exists $r>0$ such that at least $n$ points $q\in A$ satisfy $d(p,q)=r$. Say…

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

For $n$ points in the plane, let $s_i$ be the number of pairs at the $i$th distinct distance. Is $Σ_i s_i^2<n^{3+ε}$ for every fixed $ε>0$ and all sufficiently large $n$?