115 problems

For each dimension $n\ge 2$, determine whether the following implication holds: for every domain $\Omega\subset\mathbb{R}^{n}$, every $u\in W^{2,n}_{\mathrm{loc}}(\Omega)$, and eve…

Let $n$ be an integer, and let a set of points in the plane be in general position, meaning that no three points are collinear. Erdős–Szekeres convex polygon conjecture. Every set…

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 $d,r$ 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…

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 $r$ 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 $N$…

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 $F_1,\ldots,F_{d+1}$ be families of $r$ points each in $\mathds{R}^d$, considered as color classes. A colorful partition is a partition into $r$ sets $A_1,\ldots,A_r$ such that…

Let $\Omega$ and $k$ satisfy the assumptions in Section 1. Suppose also that $\Omega$ is convex and $\rho_->0$. An optimal-density uniqueness and convexity conjecture asserts that the…

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…

Hadamard-type conjecture. (i) $\mathcal G$ is concave if and only if $G_v\geq 0$ for every $x\in\Omega$. (ii) $v$ is $-\infty$-concave, meaning that $\mathcal G$ has convex level s…

Maximum colourful simplicial depth conjecture. The maximum colourful simplicial depth of any point in the interior of the core is

Let $0\leq k\leq d-1$ and let $S_0,S_1,\ldots,S_k$ be finite sets of points in $\mathbb R^d$, with … for $i=0,1,\ldots,k$. A common convex-hull transversal is a $k$-flat meeting ea…

A quadrisecant of a knot is a straight line that intersects the knot four times. It is alternating when the four intersection points alternate between the north and south poles of…

Consider a domain $U\subset{\mathbb R}P^n$ bounded by a connected smooth hypersurface $B$. Suppose that the second fundamental form of $B$ is non-degenerate at every point and has…

Let $(\Sigma,\eta)$ be a contact Hamiltonian manifold, where $\eta$ is a contact Hamiltonian structure on $\Sigma$. A heterodimensional cycle is a pair of heteroclinic trajectories…

Let $\Omega$ be a uniformly convex domain in $\mathbb{R}^{2n}$ with $n>1$, and let $c_k(\Omega)$ denote its Gutt--Hutchings capacities. The first capacity $c_1(\Omega)$ is the syst…

Let $S$ be a surface, and let $\operatorname{Convex}(S)$ denote the space of convex curve functionals on $S$, equipped with the product topology inherited from its values on the cu…

Liu–Pego's local convexity conjecture. Every such map $\phi$ must necessarily be locally convex in $\Omega$.

Let $X$ be a Banach space, let $k\ge 2$ be fixed, and let $n\ge 2$. For $n$-point sets $Q_1,\dots,Q_n\subset X$, define $\operatorname{Tv}(X,k,n)$ as the smallest radius factor suc…

Jeffs's conjecture. If $\mathcal{C}$ has up to four maximal codewords, then $\mathcal{C}$ is convex if and only if $\mathcal{C}$ has no local obstructions and no wheels.

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

Subset-normal bound conjecture. The inequalities above hold.

Facet-normal characterization conjecture. The normals of $K$ are the same as either those of a simplex bounded by an extra facet, or those of a pyramid that is not a simplex.