Hollow lattice polytopes: Latest advances in classification and relations with the width

Abstract

[Abstract]: Volume and width are two of the most studied ways to measure a body. With the concept of width, natural questions arise on lattice polytopes. A particular question someone can ask is the maximum width that a lattice polytope can have for arbitrary dimension or which is the finiteness threshold of a hollow lattice polytope. These questions are the main topic of this survey. Focusing in these concepts have helped to bring new upper bounds in the volume of hollow polytopes with lower bounded width, which lead into new advances in their classification. Some examples are maximal 3-hollow polytopes or empty 4-simplices. In this survey, we recap all background knowledge regarding this concept and get together all new results that have been published during the recent years. These advances in lattice polytopes may lead to new results in big questions of convex geometry as it is finding new examples or bounds in relation with the Flatness Theorem for convex bodies and polytopes.