In this paper, we construct an injection $A \times B \rightarrow M \times M$ from the product of any two nonempty subsets of the symmetric group into the square of their midpoint set, where the metric is that corresponding to the conjugacy class of transpositions. If $A$ and $B$ are disjoint, our construction allows to inject two copies of $A \times B$ into $M \times M$. These injections imply a positively curved Brunn-Minkowski inequality for the symmetric group analogous to that obtained by Ollivier and Villani for the hypercube. However, while Ollivier and Villani's inequality is optimal, we believe that the curvature term in our inequality can be improved. We identify a hypothetical concentration inequality in the symmetric group and prove that it yields an optimally curved Brunn-Minkowski inequality.
@article{10_37236_5429,
author = {Weerachai Neeranartvong and Jonathan Novak and Nat Sothanaphan},
title = {A curved {Brunn-Minkowski} inequality for the symmetric group},
journal = {The electronic journal of combinatorics},
year = {2016},
volume = {23},
number = {1},
doi = {10.37236/5429},
zbl = {1330.05023},
url = {http://geodesic.mathdoc.fr/articles/10.37236/5429/}
}
TY - JOUR
AU - Weerachai Neeranartvong
AU - Jonathan Novak
AU - Nat Sothanaphan
TI - A curved Brunn-Minkowski inequality for the symmetric group
JO - The electronic journal of combinatorics
PY - 2016
VL - 23
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/5429/
DO - 10.37236/5429
ID - 10_37236_5429
ER -
%0 Journal Article
%A Weerachai Neeranartvong
%A Jonathan Novak
%A Nat Sothanaphan
%T A curved Brunn-Minkowski inequality for the symmetric group
%J The electronic journal of combinatorics
%D 2016
%V 23
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/5429/
%R 10.37236/5429
%F 10_37236_5429
Weerachai Neeranartvong; Jonathan Novak; Nat Sothanaphan. A curved Brunn-Minkowski inequality for the symmetric group. The electronic journal of combinatorics, Tome 23 (2016) no. 1. doi: 10.37236/5429
