8 problems

Let $K\in\mathcal{K}^d$ be a convex body, and say that a set of directions illuminates $K$ if every point of $\operatorname{bd}K$ is illuminated by at least one direction in the se…

Let $\mathbf{K}\subset\mathbb{E}^d$ be a $1$-unconditionally symmetric cap body, meaning that $\mathbf{K}$ is symmetric about each coordinate hyperplane of $\mathbb{E}^d$, and let…

Let $\mathbf{K}$ be a $d$-dimensional convex body in $\mathbb E^d$, where $d\geq 3$, and let $I(\mathbf{K})$ denote the smallest number of directions in $\mathbb S^{d-1}$ that illu…

Let $X(\mathbf{K})$ be the smallest number of lines such that every boundary point of a convex body $\mathbf{K}$ is X-rayed along at least one of them. X-ray conjecture. For every…

Let $\mathbf{K}$ be a convex body in $\mathbb{E}^d$, $d\geq 3$, and let $\operatorname{him}(\mathbf{K})$ be its Helly dimension, defined as the least $h$ for which the correspondin…

Let $\mathbf{P}$ be a totally sewn neighbourly $4$-polytope in $\mathbb{E}^4$, and let $\mathbf{o}$ be an interior point of $\mathbf{P}$. Bisztriczky's separation conjecture. There…

Let $\mathbf{K}$ be a three-dimensional convex body of constant width. Constant-width illumination conjecture. The illumination number satisfies $I(\mathbf{K})=4$. The known genera…

Let $\mathbf{K}$ be a convex body in $\mathbb{E}^d$, $d\geq 3$, and let $\mathbf{o}$ be an interior point. A face is the intersection of $\mathbf{K}$ with a supporting hyperplane.…