7 problems

Three-dimensional flatness conjecture. No hollow convex $3$-body has width larger than $$ . That is,

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…

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 $K\in\mathcal K_o^n$ 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…

Let $K\in\mathcal K^n$ be a full-dimensional convex body and let $\Lambda\in\mathcal L^n$. Write $K_s$ for the symmetrization of $K$, and $K^\star$ and $\Lambda^\star$ for the pola…

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…