Title	Improved upper bounds on the diameter of lattice polytopes.
Creator	Deza, Antoine
Publisher	Banff International Research Station for Mathematical Innovation and Discovery
Date Issued	2016-10-27T10:00
Description	Let $D(d,k)$ denote the largest possible diameter over all polytopes which vertices are drawn from $\{ 0,1,...,k\}^d$. In 1989, Naddef showed that $D(d,1)=d$. This result was generalized in 1992 by Kleinschmidt and Onn who proved that $D(d,k) \leq kd$, before Del Pia and Michini tightened in 2016 the inequality to $D(d,k) \leq (k-1/2)d$ for $k\geq 2$. We show that $D(d,k) \leq (k-2/3)d$ for $k \geq 3$. In addition, we show that $D(4,3)=8$, which substantiates the conjecture stating that $D(d,k) \leq (k+1)d/2$, and is achieved by a Minkowski sum of lattice vectors. Based on a joint work with Lionel Pournin, Universit\'e Paris 13.
Extent	45 minutes
Subject	Mathematics; Convex and discrete geometry; Algebraic topology; Combinatorics
Type	Moving Image
File Format	video/mp4
Language	eng
Notes	Author affiliation: McMaster University
Series	BIRS Workshop Lecture Videos (Oaxaca de Juárez (Mexico))
Date Available	2017-06-08
Provider	Vancouver : University of British Columbia Library
Rights	Attribution-NonCommercial-NoDerivatives 4.0 International
DOI	10.14288/1.0348155
URI	http://hdl.handle.net/2429/61876
Affiliation	Non UBC
Peer Review Status	Unreviewed
Scholarly Level	Faculty
Rights URI	http://creativecommons.org/licenses/by-nc-nd/4.0/
Aggregated Source Repository	DSpace

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