2014/03/31 by Steven Simon · 1 citation
Mathematics · #math.AT #math.CO #math.MG
published as Geom. Dedicata, 2015, 10.1007/s10711-015-0077-5 · 11 pages
arxiv created 2015/06/20 · arxiv updated 2015/06/23
Applications of harmonic analysis on finite groups are introduced to measure partition problems, with equipartitions obtained as the vanishing of prescribed Fourier transforms. For elementary abelian groups Zpk, p an odd prime, equipartitions are by k-tuples of complex regular p-fans in ℂd, analogues of the famous Grünbaum problem on equipartitions in ℝd by k-tuples of hyperplanes (i.e., regular 2-fans). Here the number of regions is a prime power, as usual in topological applications to combinatorial geometry. For general abelian groups, however, the Fourier perspective yields new classes of equipartitions by families of complex regular fans Fq1,…, Fqk (such as those of a "Makeev-type"), including when the number of regions is not a prime power.