Abstract
In this paper we prove the conjecture of Korevaar and Meyers: for each $N\ge c_dt^d$, there exists a spherical $t$-design in the sphere $S^d$ consisting of $N$ points, where $c_d$ is a constant depending only on $d$.
-
[B]
B. Bajnok, "Construction of spherical $t$-designs," Geom. Dedicata, vol. 43, iss. 2, pp. 167-179, 1992.
@article {B, MRKEY = {1180648},
AUTHOR = {Bajnok, Bela},
TITLE = {Construction of spherical {$t$}-designs},
JOURNAL = {Geom. Dedicata},
FJOURNAL = {Geometriae Dedicata},
VOLUME = {43},
YEAR = {1992},
NUMBER = {2},
PAGES = {167--179},
ISSN = {0046-5755},
CODEN = {GEMDAT},
MRCLASS = {05B30},
MRNUMBER = {1180648},
MRREVIEWER = {J. J. Seidel},
DOI = {10.1007/BF00147866},
ZBLNUMBER = {0765.05032},
} -
[Bannai1]
E. Bannai and R. M. Damerell, "Tight spherical designs. I," J. Math. Soc. Japan, vol. 31, iss. 1, pp. 199-207, 1979.
@article {Bannai1, MRKEY = {0519045},
AUTHOR = {Bannai, E. and Damerell, R. M.},
TITLE = {Tight spherical designs. {I}},
JOURNAL = {J. Math. Soc. Japan},
FJOURNAL = {Journal of the Mathematical Society of Japan},
VOLUME = {31},
YEAR = {1979},
NUMBER = {1},
PAGES = {199--207},
ISSN = {0025-5645},
CODEN = {NISUBC},
MRCLASS = {05B30},
MRNUMBER = {0519045},
MRREVIEWER = {J. J. Seidel},
DOI = {10.2969/jmsj/03110199},
ZBLNUMBER = {0403.05022},
} -
[Bannai2]
E. Bannai and R. M. Damerell, "Tight spherical designs. II," J. London Math. Soc., vol. 21, iss. 1, pp. 13-30, 1980.
@article {Bannai2, MRKEY = {0576179},
AUTHOR = {Bannai, E. and Damerell, R. M.},
TITLE = {Tight spherical designs. {II}},
JOURNAL = {J. London Math. Soc.},
FJOURNAL = {The Journal of the London Mathematical Society. Second Series},
VOLUME = {21},
YEAR = {1980},
NUMBER = {1},
PAGES = {13--30},
ISSN = {0024-6107},
CODEN = {JLMSAK},
MRCLASS = {05B05},
MRNUMBER = {0576179},
MRREVIEWER = {J. J. Seidel},
DOI = {10.1112/jlms/s2-21.1.13},
ZBLNUMBER = {0436.05018},
} -
[BV1]
A. V. Bondarenko and M. S. Viazovska, "Spherical designs via Brouwer fixed point theorem," SIAM J. Discrete Math., vol. 24, iss. 1, pp. 207-217, 2010.
@article {BV1, MRKEY = {2600661},
AUTHOR = {Bondarenko, Andriy V. and Viazovska, Maryna S.},
TITLE = {Spherical designs via {B}rouwer fixed point theorem},
JOURNAL = {SIAM J. Discrete Math.},
FJOURNAL = {SIAM Journal on Discrete Mathematics},
VOLUME = {24},
YEAR = {2010},
NUMBER = {1},
PAGES = {207--217},
ISSN = {0895-4801},
MRCLASS = {52C35 (41A55 41A63)},
MRNUMBER = {2600661},
DOI = {10.1137/080738313},
ZBLNUMBER = {1229.05057},
} -
[BL]
J. Bourgain and J. Lindenstrauss, "Distribution of points on spheres and approximation by zonotopes," Israel J. Math., vol. 64, iss. 1, pp. 25-31, 1988.
@article {BL, MRKEY = {0981745},
AUTHOR = {Bourgain, J. and Lindenstrauss, J.},
TITLE = {Distribution of points on spheres and approximation by zonotopes},
JOURNAL = {Israel J. Math.},
FJOURNAL = {Israel Journal of Mathematics},
VOLUME = {64},
YEAR = {1988},
NUMBER = {1},
PAGES = {25--31},
ISSN = {0021-2172},
CODEN = {ISJMAP},
MRCLASS = {46B20 (52A25 52A45)},
MRNUMBER = {0981745},
MRREVIEWER = {T. Figiel},
DOI = {10.1007/BF02767366},
ZBLNUMBER = {0667.52001},
} -
[BDN]
P. Boyvalenkov, D. Danev, and S. Nikova, "Nonexistence of certain spherical designs of odd strengths and cardinalities," Discrete Comput. Geom., vol. 21, iss. 1, pp. 143-156, 1999.
@article {BDN, MRKEY = {1661307},
AUTHOR = {Boyvalenkov, P. and Danev, D. and Nikova, S.},
TITLE = {Nonexistence of certain spherical designs of odd strengths and cardinalities},
JOURNAL = {Discrete Comput. Geom.},
FJOURNAL = {Discrete \& Computational Geometry. An International Journal of Mathematics and Computer Science},
VOLUME = {21},
YEAR = {1999},
NUMBER = {1},
PAGES = {143--156},
ISSN = {0179-5376},
CODEN = {DCGEER},
MRCLASS = {05B05},
MRNUMBER = {1661307},
MRREVIEWER = {Giuseppe Pellegrino},
DOI = {10.1007/PL00009406},
ZBLNUMBER = {0921.05016},
} -
[CCEK]
H. Cohn, J. H. Conway, N. D. Elkies, and A. Kumar, "The $D_4$ root system is not universally optimal," Experiment. Math., vol. 16, iss. 3, pp. 313-320, 2007.
@article {CCEK, MRKEY = {2367321},
AUTHOR = {Cohn, Henry and Conway, John H. and Elkies, Noam D. and Kumar, Abhinav},
TITLE = {The {$D\sb 4$} root system is not universally optimal},
JOURNAL = {Experiment. Math.},
FJOURNAL = {Experimental Mathematics},
VOLUME = {16},
YEAR = {2007},
NUMBER = {3},
PAGES = {313--320},
ISSN = {1058-6458},
MRCLASS = {52C17 (05B40 52A40)},
MRNUMBER = {2367321},
DOI = {10.1080/10586458.2007.10129008},
ZBLNUMBER = {1137.05020},
} -
[CK]
H. Cohn and A. Kumar, "Universally optimal distribution of points on spheres," J. Amer. Math. Soc., vol. 20, iss. 1, pp. 99-148, 2007.
@article {CK, MRKEY = {2257398},
AUTHOR = {Cohn, Henry and Kumar, Abhinav},
TITLE = {Universally optimal distribution of points on spheres},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {20},
YEAR = {2007},
NUMBER = {1},
PAGES = {99--148},
ISSN = {0894-0347},
MRCLASS = {52A40 (41A63 52C17)},
MRNUMBER = {2257398},
MRREVIEWER = {P. McMullen},
DOI = {10.1090/S0894-0347-06-00546-7},
ZBLNUMBER = {1198.52009},
} -
[CFL]
X. Chen, A. Frommer, and B. Lang, "Computational existence proofs for spherical $t$-designs," Numer. Math., vol. 117, iss. 2, pp. 289-305, 2011.
@article {CFL, MRKEY = {2754852},
AUTHOR = {Chen, Xiaojun and Frommer, Andreas and Lang, Bruno},
TITLE = {Computational existence proofs for spherical {$t$}-designs},
JOURNAL = {Numer. Math.},
FJOURNAL = {Numerische Mathematik},
VOLUME = {117},
YEAR = {2011},
NUMBER = {2},
PAGES = {289--305},
ISSN = {0029-599X},
CODEN = {NUMMA7},
MRCLASS = {65D30 (65C05 65G20)},
MRNUMBER = {2754852},
MRREVIEWER = {Michael Gnewuch},
DOI = {10.1007/s00211-010-0332-5},
ZBLNUMBER = {1208.65032},
} -
[CW]
X. Chen and R. S. Womersley, "Existence of solutions to systems of underdetermined equations and spherical designs," SIAM J. Numer. Anal., vol. 44, iss. 6, pp. 2326-2341, 2006.
@article {CW, MRKEY = {2272596},
AUTHOR = {Chen, Xiaojun and Womersley, Robert S.},
TITLE = {Existence of solutions to systems of underdetermined equations and spherical designs},
JOURNAL = {SIAM J. Numer. Anal.},
FJOURNAL = {SIAM Journal on Numerical Analysis},
VOLUME = {44},
YEAR = {2006},
NUMBER = {6},
PAGES = {2326--2341},
ISSN = {0036-1429},
MRCLASS = {65H10 (65G20)},
MRNUMBER = {2272596},
MRREVIEWER = {Ljiljana Petkovi{ć}},
DOI = {10.1137/050626636},
ZBLNUMBER = {1129.65035},
} -
[CS] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Third ed., New York: Springer-Verlag, 1999, vol. 290.
@book {CS, MRKEY = {1662447},
AUTHOR = {Conway, J. H. and Sloane, N. J. A.},
TITLE = {Sphere Packings, Lattices and Groups},
SERIES = {Grundl. Math. Wissen.},
VOLUME = {290},
EDITION = {Third},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1999},
PAGES = {lxxiv+703},
ISBN = {0-387-98585-9},
MRCLASS = {11H31 (05B40 11H06 20D08 52C07 52C17 94B75 94C30)},
MRNUMBER = {1662447},
MRREVIEWER = {Renaud Coulangeon},
ZBLNUMBER = {0915.52003},
} -
[DGS]
P. Delsarte, J. M. Goethals, and J. J. Seidel, "Spherical codes and designs," Geometriae Dedicata, vol. 6, iss. 3, pp. 363-388, 1977.
@article {DGS, MRKEY = {0485471},
AUTHOR = {Delsarte, P. and Goethals, J. M. and Seidel, J. J.},
TITLE = {Spherical codes and designs},
JOURNAL = {Geometriae Dedicata},
FJOURNAL = {Geometriae Dedicata},
VOLUME = {6},
YEAR = {1977},
NUMBER = {3},
PAGES = {363--388},
ISSN = {0046-5755},
MRCLASS = {05B99},
MRNUMBER = {0485471},
MRREVIEWER = {Mikhail Deza},
ZBLNUMBER = {0376.05015},
DOI = {10.1007/BF03187604},
} -
[HS]
R. H. Hardin and N. J. A. Sloane, "McLaren’s improved snub cube and other new spherical designs in three dimensions," Discrete Comput. Geom., vol. 15, iss. 4, pp. 429-441, 1996.
@article {HS, MRKEY = {1384885},
AUTHOR = {Hardin, R. H. and Sloane, N. J. A.},
TITLE = {Mc{L}aren's improved snub cube and other new spherical designs in three dimensions},
JOURNAL = {Discrete Comput. Geom.},
FJOURNAL = {Discrete \& Computational Geometry. An International Journal of Mathematics and Computer Science},
VOLUME = {15},
YEAR = {1996},
NUMBER = {4},
PAGES = {429--441},
ISSN = {0179-5376},
CODEN = {DCGEER},
MRCLASS = {52B11 (05B30)},
MRNUMBER = {1384885},
MRREVIEWER = {J. J. Seidel},
DOI = {10.1007/BF02711518},
ZBLNUMBER = {0858.05024},
} -
[KM]
J. Korevaar and J. L. H. Meyers, "Spherical Faraday cage for the case of equal point charges and Chebyshev-type quadrature on the sphere," Integral Transform. Spec. Funct., vol. 1, iss. 2, pp. 105-117, 1993.
@article {KM, MRKEY = {1421438},
AUTHOR = {Korevaar, J. and Meyers, J. L. H.},
TITLE = {Spherical {F}araday cage for the case of equal point charges and {C}hebyshev-type quadrature on the sphere},
JOURNAL = {Integral Transform. Spec. Funct.},
FJOURNAL = {Integral Transforms and Special Functions. An International Journal},
VOLUME = {1},
YEAR = {1993},
NUMBER = {2},
PAGES = {105--117},
ISSN = {1065-2469},
CODEN = {ITSFEQ},
MRCLASS = {41A55 (78A30)},
MRNUMBER = {1421438},
DOI = {10.1080/10652469308819013},
ZBLNUMBER = {0823.41026},
} -
[SK2]
A. B. J. Kuijlaars and E. B. Saff, "Asymptotics for minimal discrete energy on the sphere," Trans. Amer. Math. Soc., vol. 350, iss. 2, pp. 523-538, 1998.
@article {SK2, MRKEY = {1458327},
AUTHOR = {Kuijlaars, A. B. J. and Saff, E. B.},
TITLE = {Asymptotics for minimal discrete energy on the sphere},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {350},
YEAR = {1998},
NUMBER = {2},
PAGES = {523--538},
ISSN = {0002-9947},
CODEN = {TAMTAM},
MRCLASS = {11K41 (78A30)},
MRNUMBER = {1458327},
MRREVIEWER = {M. Lawrence Glasser},
DOI = {10.1090/S0002-9947-98-02119-9},
ZBLNUMBER = {0896.52019},
} -
[MNW]
H. N. Mhaskar, F. J. Narcowich, and J. D. Ward, "Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature," Math. Comp., vol. 70, iss. 235, pp. 1113-1130, 2001.
@article {MNW, MRKEY = {1710640},
AUTHOR = {Mhaskar, H. N. and Narcowich, F. J. and Ward, J. D.},
TITLE = {Spherical {M}arcinkiewicz-{Z}ygmund inequalities and positive quadrature},
JOURNAL = {Math. Comp.},
FJOURNAL = {Mathematics of Computation},
VOLUME = {70},
YEAR = {2001},
NUMBER = {235},
PAGES = {1113--1130},
ISSN = {0025-5718},
CODEN = {MCMPAF},
MRCLASS = {41A55 (42C10 65D30)},
MRNUMBER = {1710640},
DOI = {10.1090/S0025-5718-00-01240-0},
ZBLNUMBER = {0980.76070},
} -
[M]
O. R. Musin, "The kissing number in four dimensions," Ann. of Math., vol. 168, iss. 1, pp. 1-32, 2008.
@article {M, MRKEY = {2415397},
AUTHOR = {Musin, Oleg R.},
TITLE = {The kissing number in four dimensions},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {168},
YEAR = {2008},
NUMBER = {1},
PAGES = {1--32},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {52C17 (52B40 52C07 52C15 52C45)},
MRNUMBER = {2415397},
MRREVIEWER = {Nadia Benakli},
DOI = {10.4007/annals.2008.168.1},
ZBLNUMBER = {1169.52008},
} -
[OCC]
D. O’Regan, Y. J. Cho, and Y. Chen, Topological Degree Theory and Applications, Chapman & Hall/CRC, Boca Raton, FL, 2006, vol. 10.
@book {OCC, MRKEY = {2223854},
AUTHOR = {O'Regan, Donal and Cho, Yeol Je and Chen, Yu-Qing},
TITLE = {Topological Degree Theory and Applications},
SERIES = {Ser. Math. Anal. Appl.},
VOLUME = {10},
PUBLISHER = {Chapman \& Hall/CRC, Boca Raton, FL},
YEAR = {2006},
PAGES = {iv+221},
ISBN = {1-58488-648-X; 978-1-58488-648-8},
MRCLASS = {47H11 (47H10 47H14 47N20 58C30)},
MRNUMBER = {2223854},
MRREVIEWER = {Maria Patrizia Pera},
DOI = {10.1201/9781420011487},
ZBLNUMBER = {1095.47001},
} -
[SK]
E. B. Saff and A. B. J. Kuijlaars, "Distributing many points on a sphere," Math. Intelligencer, vol. 19, iss. 1, pp. 5-11, 1997.
@article {SK, MRKEY = {1439152},
AUTHOR = {Saff, E. B. and Kuijlaars, A. B. J.},
TITLE = {Distributing many points on a sphere},
JOURNAL = {Math. Intelligencer},
FJOURNAL = {The Mathematical Intelligencer},
VOLUME = {19},
YEAR = {1997},
NUMBER = {1},
PAGES = {5--11},
ISSN = {0343-6993},
CODEN = {MAINDC},
MRCLASS = {70F10 (52C99 82B05)},
MRNUMBER = {1439152},
MRREVIEWER = {T. Erber},
DOI = {10.1007/BF03024331},
ZBLNUMBER = {0901.11028},
} -
[SZ]
P. D. Seymour and T. Zaslavsky, "Averaging sets: a generalization of mean values and spherical designs," Adv. in Math., vol. 52, iss. 3, pp. 213-240, 1984.
@article {SZ, MRKEY = {0744857},
AUTHOR = {Seymour, P. D. and Zaslavsky, Thomas},
TITLE = {Averaging sets: a generalization of mean values and spherical designs},
JOURNAL = {Adv. in Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {52},
YEAR = {1984},
NUMBER = {3},
PAGES = {213--240},
ISSN = {0001-8708},
CODEN = {ADMTA4},
MRCLASS = {05B30 (26B15)},
MRNUMBER = {0744857},
MRREVIEWER = {J. J. Seidel},
DOI = {10.1016/0001-8708(84)90022-7},
ZBLNUMBER = {0596.05012},
} -
@article {Wag, MRKEY = {1089385},
AUTHOR = {Wagner, Gerold},
TITLE = {On averaging sets},
JOURNAL = {Monatsh. Math.},
FJOURNAL = {Monatshefte für Mathematik},
VOLUME = {111},
YEAR = {1991},
NUMBER = {1},
PAGES = {69--78},
ISSN = {0026-9255},
CODEN = {MNMTA2},
MRCLASS = {65D32 (41A55)},
MRNUMBER = {1089385},
MRREVIEWER = {John Todd},
DOI = {10.1007/BF01299278},
ZBLNUMBER = {0721.65011},
} -
[Yu]
V. A. Yudin, "Lower bounds for spherical designs," Izv. Ross. Akad. Nauk Ser. Mat., vol. 61, iss. 3, pp. 213-223, 1997.
@article {Yu, MRKEY = {1478566},
AUTHOR = {Yudin, V. A.},
TITLE = {Lower bounds for spherical designs},
JOURNAL = {Izv. Ross. Akad. Nauk Ser. Mat.},
FJOURNAL = {Rossiĭskaya Akademiya Nauk. Izvestiya. Seriya Matematicheskaya},
VOLUME = {61},
YEAR = {1997},
NUMBER = {3},
PAGES = {213--223},
ISSN = {0373-2436},
MRCLASS = {05B30},
MRNUMBER = {1478566},
MRREVIEWER = {Vladimir D. Tonchev},
DOI = {10.1070/im1997v061n03ABEH000132},
ZBLNUMBER = {0890.05015},
}