11 problems

Let $\Lambda\subset\mathbb{R}^d$ be a $d$-dimensional lattice. For a convex body $K\subset\mathbb{R}^d$, define its lattice-normalized volume by … Consider the maximum of…

Let $T$ be a lattice tetrahedron with exactly $k$ interior lattice points. In a minimal direction for its lattice width, consider the consecutive lattice planes meeting $T$. Lattic…

Let $T$ be a clean lattice tetrahedron, and let $i(T)$ denote its number of interior lattice points. Its lattice width is the minimum, over nonzero integer linear forms, of the num…

A thin simplex is a lattice simplex of lattice width at most one, and its lattice width is the minimum width over all nonzero integral linear functionals. Width-one conjecture. In…

A thin simplex is a lattice simplex of lattice width at most one. Finiteness conjecture. In each even dimension there are only finitely many thin simplices of lattice width…

Let $\Lambda\subset\mathbb{R}^d$ be a full-dimensional lattice, and let $\mathrm{Flt}(d)$ be the least constant such that every convex body $C\subset\mathbb{R}^d$ with…

Let $K\in\mathcal K_o^n$ be an origin-centered convex body and let $\Lambda\in\mathcal L^n$. Write $K^\star$ and $\Lambda^\star$ for the polar body and dual lattice. Álvarez Paiva…

Linear lattice-width conjecture. The lattice width can be bounded by a function which depends only linearly on $n$. The currently known upper bound is $\mathcal{O}^{*}(n^{4/3})$, s…

Let $w(3)$ denote the maximum lattice width of a hollow convex body in dimension $3$. Codenotti–Santos conjecture. … The conjectured value is attained by a certain tetrahedron. Ave…

A hollow convex body is a three-dimensional convex body whose interior contains no lattice point. The width of a convex body is the minimum, over nonzero integer linear functionals…

Let $P$ be a convex body in $\mathbb{R}^n$ with $P \cap \mathbb{Z}^n = \emptyset$, and let $f(n)$ be a function such that $w(P) \leq f(n)$, where … is the lattice width. Linear fla…