We show that every heptagon is a section of a $3$-polytope with $6$ vertices. This implies that every $n$-gon with $n\geq 7$ can be obtained as a section of a $(2+\lfloor\frac{n}{7}\rfloor)$-dimensional polytope with at most $\lceil\frac{6n}{7}\rceil$ vertices; and provides a geometric proof of the fact that every nonnegative $n\times m$ matrix of rank $3$ has nonnegative rank not larger than $\lceil\frac{6\min(n,m)}{7}\rceil$. This result has been independently proved, algebraically, by Shitov (J. Combin. Theory Ser. A 122, 2014).
@article{10_37236_4315,
author = {Arnau Padrol and Julian Pfeifle},
title = {Polygons as sections of higher-dimensional polytopes},
journal = {The electronic journal of combinatorics},
year = {2015},
volume = {22},
number = {1},
doi = {10.37236/4315},
zbl = {1312.52008},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4315/}
}
TY - JOUR
AU - Arnau Padrol
AU - Julian Pfeifle
TI - Polygons as sections of higher-dimensional polytopes
JO - The electronic journal of combinatorics
PY - 2015
VL - 22
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/4315/
DO - 10.37236/4315
ID - 10_37236_4315
ER -
%0 Journal Article
%A Arnau Padrol
%A Julian Pfeifle
%T Polygons as sections of higher-dimensional polytopes
%J The electronic journal of combinatorics
%D 2015
%V 22
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/4315/
%R 10.37236/4315
%F 10_37236_4315
Arnau Padrol; Julian Pfeifle. Polygons as sections of higher-dimensional polytopes. The electronic journal of combinatorics, Tome 22 (2015) no. 1. doi: 10.37236/4315
