31 problems

Let $d=3$ and $n=2k$, and let $\widehat{\Sigma}_k$ denote the $k$th higher secondary polytope of the configuration. Farber's conjecture. The diameter of $\widehat{\Sigma}_k$ is … T…

Schechtman's sparsification conjecture. Is there a universal constant $C$ such that there is a zonotope $\widetilde Z$ generated by at most

Zonotope sparsification conjecture. There exists a matrix $W\in\mathbb{R}^{d\times n}$ such that

Let ${\mathcal A}$ be a hyperplane arrangement and let $Z({\mathcal A})$ denote the zonotope associated with it. A zonotope is inscribable if it has a realization whose vertices li…

Let a finite collection of lattice vectors spanning $\mathbb{R}^d$ be cosimple if it has a linear dependence whose coefficients are all non-zero and have pairwise different absolut…

Let $Z_{\mathbf v}$ be a Strong Lonely Runner Zonotope, meaning a Lonely Runner Zonotope associated with a velocity vector $\mathbf v=(v_1,\dotsc,v_n)\in\mathbb{Z}_{>0}^n$ whose en…

Let $Z=\sum_{i=1}^n[\mathbf{0},\mathbf{u}_i]\subseteq\mathbb{R}^{n-1}$ be a lattice zonotope of dimension $n-1$ with $n$ generators, whose every $n-1$ generators are linearly indep…

Let $Z$ be a zonotope of the form … and let $G$ be its $1$-skeleton. Let $i(G)$ denote the Cheeger constant, or edge-expansion, of $G$. Inverse-polynomial Cheeger conjecture for zo…

Let $Z$ be a zonotope, and let $\operatorname{rk}(Z)$ denote its rank, namely the number of summands in a representation of $Z$ as the Minkowski sum of a nondegenerate collection o…

Let $\mathbf{n}\in\mathbb{Z}^d_{>0}$ be a velocity vector, and let … Here an interior lattice point means a point $\mathbf w\in\operatorname{int}(T_{\mathbf n})\cap\mathbb{Z}^d$, a…

Let $\mathbf{n}\in\mathbb{Z}^d_{>0}$ be a velocity vector, meaning that its entries are distinct and relatively prime, and let … An interior lattice point is a point…

Let $K\subseteq\mathbb{R}^d$ be a zonotope. For symmetric convex bodies, the vector balancing constant $\operatorname{vb}(K,K)$ is the least $r\geq0$ such that every finite sequenc…

Let $A\in\mathbb{R}^{m\times d}$ with $m\geq d$ and $0<\varepsilon\leq\frac12$. Sparse $\ell_1$ representation conjecture. There exists a universal constant $C>0$ and a matrix…

Let $K\subseteq\mathbb{R}^d$ be a zonotope and let $0<\varepsilon\leq\frac12$. A zonotope is a Minkowski sum of finitely many line segments; its number of segments is the number of…

Let $x_1,x_2,\ldots,x_n\in\mathbb{R}^d$ be given with $1\leq d\leq n$. For $p\in[0,\infty]$ and $2\leq k\leq d$, consider the $k$-fold exterior products…

Strong inscribability conjecture in higher rank. Every strongly inscribable arrangement of rank $d\geq 3$ is combinatorially isomorphic to the restriction of a reflection arrangeme…

Strong inscribability conjecture. Only reflection arrangements and their restrictions are strongly inscribable.

Let $\mathcal{A}\subset\mathbb{R}^{d-1}$ be a configuration of distinct points, and let $\widehat{\Sigma}_{\mathcal{A},k}$ be the convex hull of the vectors…

Zonotope extremality conjecture. The largest possible diameter of a lattice polytope contained in the hypercube $[0,k]^d$ is always achieved by a zonotope.

An inscribed zonotope is a zonotope whose vertices lie on a common sphere. A zonotope is combinatorially equivalent to another polytope if their face lattices are isomorphic. A…

Primitive-zonotope conjecture. For all $d$ and $k$, the value $b4(d,k)$ is achieved by a lattice zonotope generated by primitive segments.

Equal-edge projection conjecture. The maximum volume of a projection of $\operatorname{C}^n$ onto a subspace is attained when the projections of all edges of the cube have the same…

Let $\mathcal{A}$ be a hyperplane arrangement, let $c_0$ be a chamber, and let $(\mathcal{A},c_0)$ be all-coherent if some, equivalently every, projection $\pi$ in the interior of…

Let $n$ be a positive integer. Let $Z\subseteq\mathbb{R}^n$ be a zonotope generated by $n+1$ vectors of $\mathbb{Z}^n$ in LGP (general linear position), and let $\boldsymbol{x}$ be…

Let $S=\{\boldsymbol{z}_1,\dotsc,\boldsymbol{z}_m\}\subseteq\mathbb{Z}^n$ and let $Z$ be the lattice zonotope generated by $S$. The set $S$ is in LGP (general linear position) when…