Equipartition of several measures

We prove several results of the following type: any $d$ measures in $\mathbb R^d$ can be partitioned simultaneously into $k$ equal parts by a convex partition (this particular result is proved independently by Pablo Sober\'on). Another example is: Any convex body in the plane can be partitioned into $q$ parts of equal areas and perimeters provided $q$ is a prime power. The above results give a partial answer to several questions posed by A. Kaneko, M. Kano, R. Nandakumar, N. Ramana Rao, and I. B\'{a}r\'{a}ny. The proofs in this paper are inspired by the generalization of the Borsuk--Ulam theorem by M. Gromov and Y. Memarian. The main tolopogical tool in proving these facts is the lemma about the cohomology of configuration spaces originated in the work of V.A. Vasil'ev. A newer version of this paper, merged with the similar paper of A. Hubard and B. Aronov is {arXiv:1306.2741}.

[1]  Karol Borsuk.  Drei Sätze über die n-dimensionale euklidische Sphäre , 1933 .

[2]  J. Tukey,et al.  Generalized “sandwich” theorems , 1942 .

[3]  H. Steinhaus,et al.  Sur la division des ensembles de l'espace par les plans et des ensembles plans par les cercles , 1945 .

[4]  Imre Bárány,et al.  On a Topological Generalization of a Theorem of Tverberg , 1981 .

[5]  R. James Milgram,et al.  On the Chern classes of the regular representations of some finite groups , 1982 .

[6]  Kenneth S. Brown,et al.  Cohomology of Groups , 1982 .

[7]  Douglas R. Woodall,et al.  Sets on Which Several Measures Agree , 1985 .

[8]  V. A. Vasil'ev,et al.  Braid group cohomologies and algorithm complexity , 1988 .

[9]  The mod $2$ equivariant cohomology algebras of configuration spaces. , 1990 .

[10]  Micha Sharir,et al.  Davenport-Schinzel sequences and their geometric applications , 1995, Handbook of Computational Geometry.

[11]  Franz Aurenhammer,et al.  Minkowski-Type Theorems and Least-Squares Clustering , 1998, Algorithmica.

[12]  Jorge Urrutia,et al.  Radial Perfect Partitions of Convex Sets in the Plane , 1998, JCDCG.

[13]  David G. Kirkpatrick,et al.  Generalizing Ham Sandwich Cuts to Equitable Subdivisions , 2000 .

[14]  Sergey Bereg,et al.  Generalizing Ham Sandwich Cuts to Equitable Subdivisions , 2000, Discret. Comput. Geom..

[15]  Kaneko,et al.  Perfect Partitions of Convex Sets in the Plane , 2002 .

[16]  Splitting,et al.  Splitting Necklaces , 2003 .

[17]  J. Matousek,et al.  Using The Borsuk-Ulam Theorem , 2007 .

[18]  M. Gromov,et al.  Isoperimetry of Waists and Concentration of Maps , 2003 .

[19]  Roman N. Karasev.  Partitions of a Polytope and Mappings of a Point Set to Facets , 2005, Discret. Comput. Geom..

[20]  F. Cohen,et al.  On the Representation Theory Associated to the Cohomology of Configuration Spaces , 2006 .

[21]  Yashar Memarian,et al.  On Gromov's Waist of the Sphere Theorem , 2009, 0911.3972.

[22]  R. Karasev.  The genus and the category of configuration spaces , 2009, 0905.0555.

[23]  I. Bárány,et al.  Equipartitioning by a convex 3-fan , 2010 .

[24]  B. Aronov,et al.  Convex Equipartitions of volume and surface area , 2010, 1010.4611.

[25]  Pablo Soberón,et al.  BALANCED CONVEX PARTITIONS OF MEASURES IN ℝ d , 2010, 1010.6191.

[26]  R NANDAKUMAR,et al.  Fair partitions of polygons: An elementary introduction , 2012 .