241 research outputs found
Convex Equipartitions via Equivariant Obstruction Theory
We describe a regular cell complex model for the configuration space
F(\R^d,n). Based on this, we use Equivariant Obstruction Theory to prove the
prime power case of the conjecture by Nandakumar and Ramana Rao that every
polygon can be partitioned into n convex parts of equal area and perimeter.Comment: Revised and improved version with extra explanations, 20 pages, 7
figures, to appear in Israel J. Mat
The ideal-valued index for a dihedral group action, and mass partition by two hyperplanes
We compute the complete Fadell-Husseini index of the 8 element dihedral group
D_8 acting on S^d \times S^d, both for F_2 and for integer coefficients. This
establishes the complete goup cohomology lower bounds for the two hyperplane
case of Gr"unbaum's 1960 mass partition problem: For which d and j can any j
arbitrary measures be cut into four equal parts each by two suitably-chosen
hyperplanes in R^d? In both cases, we find that the ideal bounds are not
stronger than previously established bounds based on one of the maximal abelian
subgroups of D_8.Comment: new version revised according to referee's comments, 44 pages, many
diagrams; a shorter version of this will appear in Topology and its
Applications (ATA 2010 proceedings
Optimal bounds for the colored Tverberg problem
We prove a "Tverberg type" multiple intersection theorem. It strengthens the
prime case of the original Tverberg theorem from 1966, as well as the
topological Tverberg theorem of Barany et al. (1980), by adding color
constraints. It also provides an improved bound for the (topological) colored
Tverberg problem of Barany & Larman (1992) that is tight in the prime case and
asymptotically optimal in the general case. The proof is based on relative
equivariant obstruction theory.Comment: 17 pages, 3 figures; revised version (February 2013), to appear in J.
European Math. Soc. (JEMS
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