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arXiv · 1812.00916

Hollow polytopes of large width

Giulia Codenotti·Francisco Santos
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Abstract

We construct a hollow lattice polytope (resp. a hollow lattice simplex) of dimension $14$ (resp.$~404$) and of width $15$ (resp.$~408$). They are the first known hollow lattice polytopes of width larger than dimension. We also construct a hollow (non-lattice) tetrahedron of width $2+\sqrt2$ and conjecture that this is the maximum width among $3$-dimensional hollow convex bodies. We show that the maximum lattice width grows (at least) additively with $d$. In particular, the constructions above imply the existence of hollow lattice polytopes (resp. hollow simplices) of arbitrarily large dimension $d$ and width $\simeq 1.14 d$ (resp.$~\simeq 1.01 d$).

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Giulia Codenotti, Francisco Santos. 2018-12-03. Hollow polytopes of large width. https://doi.org/10.1090/proc%2F14721

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