3 problems

Let $cmathcal F_1,\dots,\mathcal F_{d+1}$ be collections of volume-$1$ convex sets in $cmathbb R^d$, with $|\mathcal F_i|=n$ for every $i\in[d+1]$. A colorful homogeneous selection…

For integers $d,r\geq 1$, let $cmathcal F_1,\dots,\mathcal F_{2d}$ be families in $cmathbb R^d$, each containing $f(d,r)$ convex sets of volume $1$. A colorful Tverberg conjecture.…

Hadwiger's theorem concerns common line transversals to families of convex sets in the plane, and it has a generalization to hyperplane transversals in any dimension. A family of c…