The Brunn-Minkowski inequality and nontrivial cycles in the discrete torus

Let (C_m^d)_∞ denote the graph whose set of vertices is Z_m^d in which two distinct vertices are adjacent iff in each coordinate either they are equal or they differ, modulo m, by at most 1. Bollobás, Kindler, Leader, and O'Donnell proved that the minimum possible cardinality of a set of vertices of (C_m ^d)_∞ whose deletion destroys all topologically nontrivial cycles is m^d - (m - 1)^d. We present a short proof of this result, using the Brunn-Minkowski inequality, and also show that the bound can be achieved only by selecting a value x_i in each coordinate i, 1 ≤ i ≤ d, and by keeping only the vertices whose ith coordinate is not x_i for all i.