11 problems

Near-quadratic energy conjecture. If $A\subset\mathbb{R}$ is convex, then

Let $A\subset\mathbb{R}$ be a finite convex set, meaning that when $A=\{a_1<\dots<a_{|A|}\}$, its consecutive differences $a_{i+1}-a_i$ are strictly increasing. For finite sets, wr…

Let $A\subset\mathbb{R}$ be a finite convex set, meaning that if $A=\{a_1<a_2<\cdots<a_k\}$, then … for every $1<i<k$. Erdős's conjecture. For every $\epsilon>0$, … This conjecture…

Let $S=(\mathbb{R}^2,E)$ be an $\mathbb{R}^2$-convex society, meaning that every approval set in $E$ is a convex subset of $\mathbb{R}^2$. It is $(3,4)$-agreeable when among every…

Erdős's conjecture. For every $\epsilon>0$, there is a constant $C(\epsilon)>0$ such that every finite convex set $A$ satisfies

Let a symmetric spectral set be a state space that is spectral and symmetric, where spectrality means that every state has a common spectrum across its orthogonal decompositions an…

Let a state space be a convex set whose states admit spectral decompositions, and call it symmetric when any $n$-frame can be transformed into any other $n$-frame by an affine map…

Let $S$ be a finite-dimensional convex compact set that is spectral, meaning that all of its states have orthogonal decompositions with a common spectrum, and suppose that the symm…

Let $D\subset\mathbb R^3$ be a convex set, and call a scaled and translated copy of $D$ without rotation a homothetic copy of $D$. Let $S$ be a finite set of points in…

Let $D$ be a convex set in the plane, and call a scaled and translated copy of $D$ without rotation a homothetic copy of $D$. Let $S$ be a finite set of points. The three-color hom…

Let $D$ be a convex set in the plane, let $S$ be a finite set of points, and let $m$ be a positive integer. Pach's conjecture. For every convex set $D$ there is an $m$ such that an…