17 problems

Let $n=2^s$ with $s\geq 2$, and let $P$ be a convex small $n$-gon of maximal perimeter. Its diameter graph is the graph whose vertices are the vertices of $P$ and whose edges join…

Witsenhausen's conjecture. The maximum is attained when the points are distributed among the $n+1$ vertices of a regular simplex of edge length $1$. Witsenhausen established the up…

Let $n$ points be given in the Euclidean plane, and consider all angles formed by triples of these points. Minimum-angle conjecture. Among all angles formed by triples of $n$ point…

A convex curve of constant halving distance is a convex curve for which the distance between each pair of antipodal points, or halving points, is constant. Let $C_\triangle$ denote…

Bezdek's conjecture. A $\lambda$-convex spindle has the largest circumradius among all $\lambda$-convex bodies in $\mathbb{R}^n$ with fixed mean width.

Let $B^n$ be the unit ball in $\mathbb{R}^n$, and let $A\subseteq B^n$ be a measurable set containing no pair of points at distance $1$. Write $\mu$ for Lebesgue measure and let…

Let $G(\mathbb{R}^2,\{1\})$ be the graph whose vertices are points of $\mathbb{R}^2$, with two points adjacent when their distance is $1$, and let $\alpha_{\bar{\delta}}$ denote it…

Let $k$ great circles be arranged on a sphere, partitioning it into regions, and minimize the maximum inradius of those regions. Fejes Tóth's conjecture. In an optimal arrangement,…

Let $c(d,n,k)$ denote the extremal signed-sum quantity studied in the paper, and let $\omega_{d+1}$ be the corresponding extremal vector configuration. For a linear subspace…

A convex surface $S$ is the boundary of a convex body in $\mathbb{R}^3$, or a doubly covered planar convex body. Its intrinsic diameter is … where $\rho(x,y)$ is the length of a sh…

Let $K$ be a convex pentagon in the plane, and let $K_1$ be the pentagon bounded by the diagonals of $K$. Affine regular pentagon conjecture. The maximum of the ratio between the a…

Optimal-cap conjecture. These bounds are all equalities. In particular, for $t\leq-1/2$, forbidding only the single inner product $t$ suffices to attain the same bound as the corre…

Let $d\ge 2$ and $n\ge 2$. Let $f_d(n)$ denote the largest volume of a $d$-dimensional open set with $n$ connected components and no pair of points at an integral distance. Let…

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…

Let $K$ be a convex body in $\mathbb{R}^d$, let $T^d$ be a $d$-dimensional simplex, let $p\geqslant 1$, and let $n\geqslant d+1$. Write $\mathbb{E}^p_*(K)$ and $\mathbb{E}^p_n(K)$…

Near-linear upper-bound conjecture. The function $S_{P}^{\nparallel}(n)$ should satisfy this bound for every parallelogram-free pattern $P$. The paper suggests that the constructio…

Equilateral-triangle side-length conjecture. One has