10 problems

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…

A convexity space is a pair $(X,\mathcal{C})$ such that $\varnothing,X\in\mathcal{C}$, the family $\mathcal{C}$ is closed under arbitrary intersections, and the union of every incl…

Quadratic-asymptotics conjecture. The function $m(n)$ has quadratic asymptotics in $n$.

Colorful Tverberg conjecture. There exists a colorful partition $\\{P_1,\dots,P_k\\}$ such that the convex hulls of $P_1,\dots,P_k$ intersect.

Calder–Eckhoff partition conjecture. The inequality

Sierksma's Dutch cheese conjecture. There are at least $((r-1)!)^d$ partitions of $X$ into parts whose convex hulls intersect. The conjecture extends the Dutch cheese conjecture fr…

Let a depth measure have Tverberg depth regions, and let the cascade condition refer to the sum of the dimensions of its depth regions across depth levels. Weak Cascade Conjecture.…

Classification conjecture. For every Tverberg type $\mathcal I$, if $\mathcal I$ is colorful, then $\operatorname{Tv}_{\mathcal I}$ is unavoidable; otherwise,…

Core decomposition conjecture. There are $m$ disjoint subsets $\boldsymbol{\mathbf{S}}_1,\dots,\boldsymbol{\mathbf{S}}_m$ of $\boldsymbol{\mathbf{A}}$, with

Let $q\geq 2$ be a prime power, $d\geq 1$, and let $G$ be a graph on $(d+1)(q-1)+1$ vertices. A graph is a $(d,q)$--Tverberg graph if it has the corresponding Tverberg property for…