4 problems

Let $d$ be a positive integer, and let ${\mathcal F}$ be a family of at least two pairwise disjoint open convex sets in ${\mathbb R}^d$. A line in ${\mathbb R}^d$ is transversal to…

Let $\mathcal{K}$ be a finite family of pairwise intersecting convex sets in $\mathbb{R}^3$. Constant-line transversal conjecture. The family $\mathcal{K}$ admits a transversal by…

Let $\mathcal{K}$ be a finite family of $n$ convex sets in $\mathbb{R}^3$, and suppose that every two members $K_1,K_2\in\mathcal{K}$ satisfy $K_1\cap K_2\neq\emptyset$. Fractional…

Let $\mathcal{K}=\mathcal{K}_1\uplus\mathcal{K}_2$ be a finite 2-colored family of convex sets in $\mathbb{R}^3$. Suppose that every $K_1\in\mathcal{K}_1$ and $K_2\in\mathcal{K}_2$…