263 problems

Let $r,d$ be positive integers, and let a Tverberg partition of a set of points be a partition into $r$ parts whose convex hulls intersect. For a set of $(r-1)(d+1)+1$ points in…

Hadwiger–Debrunner's conjecture. For every $p\geqslant q\geqslant d+1$, there exists a constant $c=c(p,q;d)$ such that every family of compact, convex sets in $\mathbb{R}^d$ with t…

For integers $d$, $r$, and $k$ with $2\le k\le r$, let $T(d,r,k)$ be the smallest integer $n$ such that every set of $n$ points in $\mathbb R^d$ can be partitioned into $r$ parts w…

Let $p\in\mathbb{R}[\mathbf{x}]$ be a real-zero (RZ) polynomial with $p(\mathbf{0})=1$. For a polynomial $f$, write $\operatorname{rcs}(f)$ for its rigidly convex set. Generalized…

Let $d+1$ sets, called colours, each consist of $d+1$ points in general position in $\mathbb{R}^d$, and let $p$ be any point in the convex hull of each set. Colourful simplicial de…

Given $d+1$ color classes in $\mathbb R^d$, each of size $r$, a rainbow set contains at most one point from each color class. Bárány–Larman's colored Tverberg conjecture. One can p…

Let $0 \le k < d$ be integers, and let $r_0, \dots, r_k$ be positive integers. Suppose that $P_0, \dots, P_k$ are sets of points in $\mathds{R}^d$ satisfying … for each $i$. Tverbe…

Let $Q_1,\dots,Q_{d+1}$ be pairwise disjoint sets of $k$ points each in $\mathbb{R}^d$. A transversal is a subset of their union containing exactly one element from each $Q_i$. Col…

Dol'nikov's conjecture. There exists $j\in[3]$ such that $\mathcal{F}_j$ can be pierced by $3$ points.

White's conjecture. The only convex eternal solutions are the translators.

Let $d,r$ be positive integers and let $k$ be a non-negative integer. A Tverberg $(r+1)$-partition of a finite point set is a partition into $r+1$ parts whose convex hulls have a c…

Peeling-sequence growth conjecture. The minimal number of peeling sequences of $n$ points in $\mathbb{R}^d$ is at least

Choi–Kribs–Życzkowski's convexity conjecture. Higher-rank numerical ranges $\Lambda_k(A)$ are always convex. This conjecture was proved by Woerdeman in 2008, so the claim is solved…

Let $\mathcal{F}$ be a finite family of convex sets in $\mathbb{R}^2$. The family has the $T(3)$ property if every three sets of $\mathcal{F}$ can be pierced by a line, and its lin…

Convexity conjecture. For every integer $n\geqslant 2$,

Let $n(d,r)$ be the smallest integer such that, for every map $f:\Delta_{n-1}\rightarrow\mathbb{R}^{d}$ and every coloring of the vertices of $\Delta_{n-1}$ by $d+1$ colors, with e…

Let $r$ and $d$ be positive integers, and let $N$ denote the number of points under consideration. Birch's conjecture. Any $(r-1)(d+1)+1$ points in $\mathbb{R}^{d}$ can be partitio…

Let $r,d$ be fixed positive integers, let $N$ be a positive integer, and let $p_r$ be the probability that a random permutation of $r$ numbers has at least one fixed point. For $N$…

Audenaert–Datta conjecture. If $1\leq p\leq 2$, $-1\leq q<0$, and $(p,q)\ne(1,-1)$, then $\Psi_{p,q,1/(p+q)}(A,B)$ is jointly convex in $(A,B)$. This conjecture concerns the joint…

Holmsen–Kynčl–Valculescu conjecture. If $X$ has a partition into $n$ subsets of size $\ell$ such that each subset contains points of at least $d$ colors, then $X$ has such a partit…

Let $V$ be a subvariety of the torus $(\mathbb{C}^\times)^n$ of codimension $k{+}1$, and let its amoeba be the image under the coordinatewise map $z\mapsto \log|z|$. A subset…

Füredi–Lagarias–Morgan conjecture. There exists an $\epsilon>0$ such that every equilateral set in $M$ satisfies

Poisson convexity conjecture. The image

Let $\Omega$ be a convex domain, and let $D=D_{\alpha,A}$ be an optimal configuration in $\Omega$. Write $D^c$ for its complement and let $\overline{\alpha}_\Omega(A)$ denote the t…

Let $w_2(\ell_{\infty}^n)$ denote the second Whitney constant associated with the $n$-dimensional cube $Q^n$, and let $E_2(f;Q^n)$ and $\omega_2(f:Q^n)$ denote the corresponding qu…