7 problems

Let $A$subset be a convex body in $\mathbb{R}^n$, and let $2\leq r\leq n$. For convex bodies $B_1,\dots,B_r$, write $V(B_1,\dots,B_r,A[n-r])$ for their mixed volume, with $A$ repea…

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 $n\ge 3$, and consider simplices inscribed into the $n$-dimensional unit ball. For a convex body $K$ in Euclidean space, its mean width is denoted by $w(K)$. Simplex mean width…

Let two unit simplices in $\mathbb R^d$ be given, one with $d+1$ vertices and the other with $\lfloor(d+1)/2\rfloor+1$ vertices. Two-unit-simplex diameter conjecture. Either the tw…

Let $a_1,\ldots,a_d$ and $b_1,\ldots,b_d$ be two simplices on $d$ vertices in $\mathbb R^d$, with $d\geq 3$, and suppose that every edge of each simplex has length at least $1$. Mo…

A unit regular simplex on $d$ vertices in $\mathbb R^d$ is a regular simplex whose edges have length $1$. Let two such simplices be given. Morić–Pach's conjecture. Any two unit reg…

Limiting domination number conjecture. As $n\to\infty$, the domination number satisfies