Measure partitions via Fourier analysis. II: Center transversality in the \({{L}^{2}}\)-norm for complex hyperplanes. (English) Zbl 1362.43005
Summary: Applications of harmonic analysis on finite groups were recently introduced to measure partition problems, with a variety of equipartition types by convex fundamental domains obtained as the vanishing of prescribed Fourier transforms. Considering the circle group, we extend this approach to the compact Lie group setting, in which case the annihilation of transforms in the classical Fourier series produces measure transversality similar in spirit to the classical centerpoint theorem of Rado: for any \({q\geq 2}\), the existence of a complex hyperplane whose surrounding regular \(q\)-fans are close – in an \({L^2}\)-sense – to equipartitioning a given set of measures. The proofs of these results represent the first application of continuous as opposed to finite group actions in the usual equivariant topological reductions prevalent in combinatorial geometry.
For Part I see [Geom. Dedicata 179, 217–228 (2015; Zbl 1348.43009)].
For Part I see [Geom. Dedicata 179, 217–228 (2015; Zbl 1348.43009)].
MSC:
| 43A50 | Convergence of Fourier series and of inverse transforms |
| 52A35 | Helly-type theorems and geometric transversal theory |
| 52A38 | Length, area, volume and convex sets (aspects of convex geometry) |
| 51M20 | Polyhedra and polytopes; regular figures, division of spaces |
| 37F20 | Combinatorics and topology in relation with holomorphic dynamical systems |
| 42A16 | Fourier coefficients, Fourier series of functions with special properties, special Fourier series |
| 28A75 | Length, area, volume, other geometric measure theory |
Citations:
Zbl 1348.43009References:
| [1] | Bárány I., Matoušek J.: Equipartition of two measures by a 4-fan. Discret. Comput. Geom. 27, 293-301 (2002) · Zbl 1002.60006 · doi:10.1007/s00454-001-0071-6 |
| [2] | Bárány I., Matoušek J.: Simultaneous partitions of measures by k-fans. Discret. Comput. Geom. 25, 319-334 (2001) · Zbl 0989.52009 · doi:10.1007/s00454-001-0003-5 |
| [3] | Blagojević, P., Frick, F., Hasse, A., Ziegler, G.: Topology of the Grünbaum-Hadwiger-Ramos hyperplane mass partition problem. arXiv:1502.02975 [math.AT] · Zbl 1396.52011 |
| [4] | Blagojević P., Karasev R.: Extensions of theorems of Rattray and Makeev. Topol. Methods Nonlinear Anal. 40, 189-213 (2012) · Zbl 1282.55008 |
| [5] | Blagojević P., Ziegler G.: Convex equipartitions via equivariant obstruction theory. Isr. J. Math. 200, 49-77 (2014) · Zbl 1305.52005 · doi:10.1007/s11856-014-1006-6 |
| [6] | Blagojević P., Ziegler G.: The ideal-valued index for a dihedral group action and mass partition by two hyperplanes. Topol. Appl. 158(12), 1326-1351 (2011) · Zbl 1225.55002 · doi:10.1016/j.topol.2011.05.008 |
| [7] | Ceccherini-Silberstein, T., Scarabotti, F., Tolli, F.: Harmonic analysis on inite groups: representation theory, Gelfand paris, and Markov chains. In: Cambridge Studies in Advanced Mathematics, vol. 108. Cambridge University Press, Cambridge (2008) · Zbl 1149.43001 |
| [8] | Erickson J., Hurtado F., Mortin P.: Centerpoint theorems for wedges. Discret. Math. Theor. Comput. Sci. 11(1), 45-54 (2009) · Zbl 1194.51003 |
| [9] | Fadell E., Husseini S.: An ideal-valued cohomological index theory with applications to Borsuk-Ulam and Bourgin-Yang theorems. Ergod. Theory Dyn. Syst. 8, 73-85 (1988) · Zbl 0657.55002 · doi:10.1017/S0143385700009342 |
| [10] | Grünbaum B.: Partitions of mass-distributions and convex bodies by hyperplanes. Pac. J. Math. 10, 1257-1261 (1960) · Zbl 0101.14603 · doi:10.2140/pjm.1960.10.1257 |
| [11] | Hadwiger H.: Simultane Vierteilung Zweier Köper. Arch. Math. (Basel) 17, 274-278 (1966) · Zbl 0137.41501 · doi:10.1007/BF01899586 |
| [12] | Hatcher. A.: Algebraic Topology. Cambridge University Press, Cambridge (2002) · Zbl 1044.55001 |
| [13] | Husemoller D.: Fiber Bundles. Springer, Berlin (1994) · doi:10.1007/978-1-4757-2261-1 |
| [14] | Karasev R.: Equipartition of a measure by \[{Z_p^k}\] Zpk-invariant fans. Discret. Comput. Geom. 43, 477-481 (2010) · Zbl 1208.52020 · doi:10.1007/s00454-009-9138-6 |
| [15] | Karasev R., Hubard A., Aronov B.: Convex equipartitions: the spicy chicken theorem. Geom. Dedicata 170, 263-279 (2014) · Zbl 1301.52010 · doi:10.1007/s10711-013-9879-5 |
| [16] | Makeev V.V.: Equipartition of a continuous mass distribution. J. Math. Sci. 140(10), 551-557 (2007) · Zbl 1151.52304 · doi:10.1007/s10958-007-0437-2 |
| [17] | Mani-Levitska P., Vrećica S., Živaljević R.: Topology and combinatorics of partitions of masses by hyperplanes. Adv. Math. 207, 266-296 (2006) · Zbl 1110.68155 · doi:10.1016/j.aim.2005.11.013 |
| [18] | Matoušek J.: Using the Borsuk-Ulam Theorem. Lectures on Topological Methods in Combinatorics and Geometry. Springer, Berlin (2008) · Zbl 1234.05002 |
| [19] | Milnor J.W., Stasheff J.D.: Characteristic Classes. Princeton University Press, NJ (1974) · Zbl 0298.57008 · doi:10.1515/9781400881826 |
| [20] | Rado R.: A theorem on general measure. J. Lond. Math. Soc. 21, 291-300 (1947) · Zbl 0061.09606 |
| [21] | Ramos R.: Equipartition of mass distributions by hyperplanes. Discret. Comput. Geom. 15, 147-167 (1996) · Zbl 0843.68120 · doi:10.1007/BF02717729 |
| [22] | Sarkaria K.S.: Tverberg partitions and Borsuk-Ulam theorems. Pac. J. Math. 196(1), 231-241 (2000) · Zbl 1045.55002 · doi:10.2140/pjm.2000.196.231 |
| [23] | Simon, S.: Measure equipartitions via finite Fourier analysis. Geom. Dedicata (2015). doi:10.1007/s10711-015-0077-5 · Zbl 1348.43009 |
| [24] | Steinhaus H.: A note on the ham sandwich theorem. Mathesis Polska 9, 26-28 (1938) |
| [25] | tom Dieck, T.: Transformation Groups. Walter de Gruyter (1987) · Zbl 0611.57002 |
| [26] | Vrećica S.T., Živaljević R.T.: Conical equipartitions of mass distributions. Discret. Comput. Geom. 25, 335-350 (2001) · Zbl 0989.52010 · doi:10.1007/s00454-001-0002-6 |
| [27] | Yao F., Dobkin D., Edelsbrunner H., Paterson M.: Partitioning space for range queries. SIAM J. Comput. 18(2), 371-384 (1989) · Zbl 0675.68066 · doi:10.1137/0218025 |
| [28] | Živaljević R.T.: Computational topology of equipartitions by hyperplanes. Topol. Methods Nonlinear Anal. 45(1), 63-90 (2015) · Zbl 1361.52008 · doi:10.12775/TMNA.2015.004 |
| [29] | Živaljević, R.T.: Topological methods. In: Goodman, J.E., O’Rourke, J. (eds.) Chapter 14 in Handbook of Discrete and Computational Geometry, pp. 305-330. Chapman & Hall/CRC (2004) · Zbl 0777.57011 |
| [30] | Živaljević R.T.: User’s guide to equivariant methods in combinatorics II. Publ. Inst. Math. Belgrade 64(78), 107-132 (1998) · Zbl 0999.52005 |
| [31] | Živaljević R.T., Vrećica S.T.: The ham sandwich theorem revisited. Isr. J. Math. 78, 21-32 (1992) · Zbl 0777.57011 · doi:10.1007/BF02801568 |
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