25 problems

Let $K\subset\mathbb R^2$ be compact. The logarithmic capacity $\operatorname{Cap}_0(K)$ and Newtonian capacity $\operatorname{Cap}_1(K)$ are the corresponding Riesz capacities. Pó…

A planar cluster encloses and separates $n$ regions of prescribed areas, with boundaries consisting of circular arcs meeting three at a time at $120$ degrees. Connectivity conjectu…

Starting from a planar tropical caustic, consider the connected components with infinitely many sides in the complement to the original caustic, and iterate the caustic constructio…

Union-of-Reuleaux-frames diameter conjecture. Then

Let $C$ be a closed equilateral polygonal chain in the Euclidean plane, and let its diameter be the supremum of distances between points of $C$. Reuleaux-frame conjecture. If the d…

Quantitative Besicovitch conjecture. There exist positive constants $\varepsilon$ and $\delta$ such that, whenever $E\subset\mathbb R^2$ is compact and

Let $P$ be a finite set of points determining $O(N)$ distinct directions, where $N=|P|$. A quadric is a plane quadric, not necessarily irreducible; in particular, three collinear p…

Let $P$subseteq$\mathbb R^2$ be a non-collinear set of $N$ points. An angle is formed by a triple of points of $P$, and the two examples described above are a regular polygon toget…

For each positive integer $n$, let … where $T^n_\sigma$ is the smallest-area unrestricted trapezoid associated with a permutation of $n$ parallelograms. Kaczynski's monotonicity co…

For the smallest unrestricted trapezoid area … where $T^n_\sigma$ is the union of the parallelograms associated with $\sigma$, Kaczynski's logarithmic-area conjecture. there exist…

Kaczynski's vanishing-area conjecture.

Consider the improper planar partitioning problem with one finite-measure chamber of prescribed area and three infinite-measure chambers, with locally perimeter minimizing understo…

Consider the planar improper partitioning problem with one finite-measure chamber of prescribed area and two infinite-measure chambers, whose interiors are pairwise disjoint and wh…

Let $\Omega$ range over planar pentagons with a fixed area, and let $T(\Omega)$ denote their Torsional rigidity. Saint-Venant's conjecture. Among all pentagons of given area, the r…

Let $E$ and $F$ be Borel sets in the plane with $\dim_{\mathcal{H}}E>0$ and $\dim_{\mathcal{H}}F>0$, and suppose that $F$ is not contained in a line. For $y$ in the plane, let…

Fix $d\geq 1$. Consider packings of unit circles in ${\Bbb R}^2$ consisting of parallel strings of unit circles whose centres are equally spaced on a straight line at distance $2d$…

Let $\gamma$ be an unbounded one-sided curve in the Euclidean plane, and let $\Omega$ be a simply connected, open, bounded planar domain. Reroot $\gamma$ at a root chosen in…

A tangent graph of a finite collection of circles in the plane has one vertex for each circle and an edge when two circles are tangent. Assume that no more than two circles pass th…

A tangent graph has one vertex for each disc and an edge when the corresponding discs are tangent. Pinchasi–Sharir tangent-incidence conjecture. (i) Planar tangent graphs with $n$…

Let $m_1(\mathbb{R}^2)$ denote the supremum of the upper densities of measurable subsets of the plane containing no pair of points at distance $1$. Erdős's conjecture. … This is a…

Let $S\subseteq\mathbb{R}^2$ be a measurable set, and let $\operatorname{b}(S)$ denote its Beer index and $\lambda$ planar Lebesgue measure. Boundary convexity conjecture. For ever…

Call a set $S\subseteq\mathbb{R}^2$ p-componentwise simply $\Delta$-connected if, for every triangle $T$ with $\partial T\subseteq S$, one has $T\subseteq S$. Assume that…

Let $A$ and $B$ be finite non-collinear point sets in $\mathbb{R}^2$. For a finite non-collinear set $S$, let $b_S$ and $i_S$ denote the numbers of points of $S$ on the boundary an…

Let an exceptional domain in $\mathbb{R}^2$ be a domain supporting the overdetermined potential-theoretic problem considered in the paper, and let its genus be finite. The three ex…

Consider four particles with arbitrary positive masses forming a convex planar central configuration, with the particles ordered as in the paper's convex quadrilateral setup. Uniqu…