3 problems

Let $cmathcal F$ be a finite family of convex sets in $cmathbb R^d$ such that the intersection of every $2d$ or fewer members has volume at least $1$. The conjecture asserts that ……

Let $r(d)$ denote the quantitative radius in the quantitative Steinitz theorem: for every convex polytope $Q\subset\mathbb{R}^d$ containing the Euclidean unit ball, at most $2d$ ve…

Let $\mathcal{F}$ be a family of convex sets in $\mathbb{R}^d$, and suppose that an $\alpha$-fraction of the $2d$-tuples of members of $\mathcal{F}$ have intersection of volume at…