2 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…

Let $X$ be a Banach space, and let $H(X)$ denote the collection of all nonempty, closed, bounded, convex subsets of $X$, equipped with the Hausdorff metric. A selector is a map…