20 problems

Bialostocki–Dierker–Voxman conjecture. The value $h_{mod}(n,q)$ exists for all $n\geqslant 3$ and $q\geqslant 2$. This is a modular variant of the Erdős–Szekeres convex polygon pro…

Strict descent conjecture. Every convex polygon satisfies the strict descent condition. The condition is introduced to construct electrostatic skeletons where different Schwarz ref…

A precompact domain $\Omega \subseteq \mathbb{R}^2$ has an electrostatic skeleton, meaning a positive measure inside $\Omega$, supported on a set with no simple loops, which genera…

Let $\mathcal{C}_{N}$ be the class of convex polygons in $\mathbb{R}^2$ with at most $N$ sides and perimeter $2\pi$, and let $\Omega^*$ be a maximizer of the first nonzero Steklov…

Existence and non-degeneracy conjecture. The supremum is attained by a convex polygon $\Omega^*$ with exactly $N$ sides. In particular, the maximizer cannot be a degenerate polygon…

Multiple-tiling conjecture. Convex pentagonal monotiles belonging to the Type 7–13 families, but not contained in $T_1$ or $T_2$, are unlikely to generate multiple types of $k$-iso…

Let $k$ be a constant, let $\alpha$ satisfy $\alpha\leq k-1$, and let $\Delta_k'(n)$ denote the normalized extremal area parameter for convex $k$-gons defined in the paper. The nor…

Let $\Delta_4'(n)$ denote the minimum, over configurations of $n$ points in the normalized setting considered in the paper, of the area of a convex quadrilateral containing four po…

Convex-pentagon ASP uniqueness conjecture. In $S_t(\delta,\epsilon)$, Figure 06AD is the only case in which the ASP comprises four types of convex pentagons; when the anterior and…

Three-polygon ASP conjecture. If

Pre-compactness conjecture. The sequence $\{P^{\ell}\}$ is pre-compact modulo affine transformations. That is, this sequence has a convergent subsequence which converges to another…

Let $\Gamma_n$ denote the set of convex $n$-gons with each edge of length $1$, and let $E_n^\alpha$ be the energy associated with the increasing continuous function…

Danzer's conjecture. If the number of tiles is $N=5$, then $P$ must be a rectangle.

Odd-tile dissection conjecture. If $N$ is an odd number with $N\geq 3$, then $P$ must be a rectangle.

Let $S$ be a finite set of points in the plane in general position, and let $k\geq 3$ and $\ell\geq 1$ be integers. For a convex $k$-gon determined by points of $S$, consider the n…

Nonexistence conjecture.

Grid obstruction conjecture. There exists a $\gamma\in(1,2)$ such that the Erdős–Szekeres construction cannot be realized in an $n^\gamma\times n^\gamma$ integer grid. The paper co…

Integer-grid problem conjecture. The answer to this problem is no for all $\gamma\ge 1$; equivalently, for every $\gamma\ge 1$, there are arbitrarily large such point sets containi…

Let a convex $n$-gon be given, and consider the distances between pairs of its vertices. The maximum number of pairs at unit distance is the maximum number of unit distances determ…

Let a convex $n$-gon be given. An isosceles triangle is a triangle formed by three of its vertices with at least two equal side lengths. The convex-polygon isosceles-triangle conje…