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Empty simplices of large width

  • Francisco Santos (Universidad de Cantabria, Santander, Spain)
Live Stream MPI für Mathematik in den Naturwissenschaften Leipzig (Live Stream)

Abstract

An empty simplex is a lattice simplex in which vertices are the only lattice points. After reviewing what is known about width of lattice polytopes and convex bodies, we show two constructions leading to the first known empty simplices of width larger than their dimension:

(1) We introduce cyclotomic simplices and exhaustively compute all the cyclotomic simplices of dimension $10$ and volume up to $2^{31}$. Among them we find five empty ones of width $11$, and none of larger width.
(2) Using circulant matrices of a very specific form, we construct empty simplices of arbitrary dimension $d$ and of width growing asymptotically as $d/$arcsinh$(1) \sim 1.1346\,d$.

The width in part (2) is (asymptotically) only $3\%$ lower than the widest convex bodies known.

This is joint work with Joseph Doolittle, Lukas Katthän, and Benjamin Nill.

Links

conference
4/6/21 4/9/21

(Polytop)ics: Recent advances on polytopes

MPI für Mathematik in den Naturwissenschaften Leipzig Live Stream

Saskia Gutzschebauch

Max Planck Institute for Mathematics in the Sciences Contact via Mail

Federico Castillo

Max Planck Institute for Mathematics in the Sciences

Giulia Codenotti

Goethe University Frankfurt

Benjamin Schröter

Royal Institute of Technology (KTH)