18 problems

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 Euclidean space $\mathbb{E}^d$. A smaller positive homothetic copy of $\mathbf{K}$ is a set of the form…

Let $n\geq 3$. For an $n$-dimensional convex body $K$, let $I(K)$ be the smallest number of directions that illuminate every point of the boundary of $K$, and let $H(n)$ be the max…

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 $K\in\mathcal{K}^d$ 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 $K$ be an $n$-dimensional convex body. A weighted collection of light sources illuminates $K$ if every boundary point is illuminated by sources whose total weight is at least…

Let $K$ be an $n$-dimensional convex body. A boundary point $x$ is illuminated from a unit direction $u$ if the ray from $x$ in direction $u$ intersects the interior of $K$, and le…

Let $C$ be a compact convex body in an $n$-dimensional vector space. For $z\in\partial C$, a vector $v$ illuminates $z$ if $z+\lambda v\in C^\circ$ for all sufficiently small…

Let $B$ be a convex body, meaning a compact convex set with non-empty interior, in ${\mathbb R}^n$, and let ${\mathcal I}(B)$ be the smallest number of non-zero directions that ill…

Let $B$ be an $n$-dimensional convex body, and let ${\mathcal I}(B)$ denote the smallest number of directions, or equivalently light sources, that illuminate $B$. A parallelotope h…

Fractional Illumination Conjecture. One has

Let $\mathbf{K}$ be an $\mathbf{o}$-symmetric $d$-dimensional convex body, and let $\operatorname{ill}(\mathbf{K})$ be its illumination parameter. Quantitative illumination conject…

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 $I_l(\mathbf{K})$ denote the minimum number of $l$-dimensional open great hemispheres of the unit sphere that illuminate a convex body $\mathbf{K}$, equivalently the associated…

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.…