 
Summary: The BrunnMinkowski Inequality and nontrivial cycles in the
discrete torus
Noga Alon
Ohad N. Feldheim
March 21, 2010
Abstract
Let (Cd
m) denote the graph whose set of vertices is Zd
m in which two distinct vertices are
adjacent iff in each coordinate they are either equal or differ, modulo m, by at most 1. Bollob´as,
Kindler, Leader and O'Donnell proved that the minimum possible cardinality of a set of vertices
of (Cd
m) whose deletion destroys all topologically nontrivial cycles is md
 (m  1)d
. We present
a short proof of this result, using the BrunnMinkowski Inequality, and also show that the bound
can only be achieved by selecting a value xi in each coordinate i, 1 i d, and by keeping only
the vertices whose ith coordinate is not xi, for all i.
Keywords: BrunnMinkowski, Discrete torus, Nontrivial cycles.
AMS classification: 05C35,05C38,05C40.
