New Upper Bounds for the Density of Translative Packings of Three-Dimensional Convex Bodies with Tetrahedral Symmetry

[1]  Henry Cohn,et al.  The sphere packing problem in dimension 8The sphere packing problem in dimension 8 , 2016, 1603.04246.

[2]  Henry Cohn,et al.  Some properties of optimal functions for sphere packing in dimensions 8 and 24 , 2016, 1603.04759.

[3]  M. Viazovska.  The sphere packing problem in dimension 8The sphere packing problem in dimension 8 , 2016, 1603.04246.

[4]  Ferenc Fodor,et al.  The Packing Density of the $$n$$n-Dimensional Cross-Polytope , 2015, Discret. Comput. Geom..

[5]  F. M. D. O. Filho,et al.  Computing Upper Bounds for the Packing Density of Congruent Copies of a Convex Body , 2013, 1308.4893.

[6]  Pablo A. Parrilo,et al.  Convex algebraic geometry and semidefinite optimization , 2013, ISSAC '13.

[7]  Yufei Zhao,et al.  SPHERE PACKING BOUNDS VIA SPHERICAL CODES , 2012, 1212.5966.

[8]  Rekha R. Thomas,et al.  Semidefinite Optimization and Convex Algebraic Geometry , 2012 .

[9]  J. Lagarias,et al.  Mysteries in packing regular tetrahedra , 2012 .

[10]  C. Zong.  On the translative packing densities of tetrahedra and cubooctahedra , 2012, 1208.0420.

[11]  F. Vallentin,et al.  Upper bounds for packings of spheres of several radii , 2012, Forum of Mathematics, Sigma.

[12]  F. Stillinger,et al.  Erratum: Optimal packings of superballs [Phys. Rev. E 79, 041309 (2009)] , 2011 .

[13]  Marjolein Dijkstra,et al.  Phase diagram of colloidal hard superballs: from cubes via spheres to octahedra , 2011, 1111.4357.

[14]  S. Torquato,et al.  Communication: a packing of truncated tetrahedra that nearly fills all of space and its melting properties. , 2011, The Journal of chemical physics.

[15]  P. Damasceno,et al.  Crystalline assemblies and densest packings of a family of truncated tetrahedra and the role of directional entropic forces. , 2011, ACS nano.

[16]  M. Dijkstra,et al.  Dense regular packings of irregular nonconvex particles. , 2011, Physical review letters.

[17]  David J. Pine,et al.  Cubic crystals from cubic colloids , 2011 .

[18]  John P. D'Angelo,et al.  Hermitian analogues of Hilbert's 17-th problem , 2010, 1012.2479.

[19]  Veit Elser,et al.  Upper Bound on the Packing Density of Regular Tetrahedra and Octahedra , 2010, Discret. Comput. Geom..

[20]  Alexander Schrijver,et al.  Invariant Semidefinite Programs , 2010, 1007.2905.

[21]  Michael Engel,et al.  Dense Crystalline Dimer Packings of Regular Tetrahedra , 2010, Discret. Comput. Geom..

[22]  S Torquato,et al.  Dense packings of polyhedra: Platonic and Archimedean solids. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  F. Stillinger,et al.  Optimal packings of superballs. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Bernd Sturmfels,et al.  Algorithms in Invariant Theory (Texts and Monographs in Symbolic Computation) , 2008 .

[25]  Henry Cohn,et al.  Universally optimal distribution of points on spheres , 2006, math/0607446.

[26]  Fabrice Rouillier,et al.  Motivations for an Arbitrary Precision Interval Arithmetic and the MPFI Library , 2005, Reliab. Comput..

[27]  P. Parrilo,et al.  Symmetry groups, semidefinite programs, and sums of squares , 2002, math/0211450.

[28]  Henry Cohn,et al.  New upper bounds on sphere packings I , 2001, math/0110009.

[29]  M. Henk,et al.  Densest lattice packings of 3-polytopes , 1999, Comput. Geom..

[30]  T. Hales.  The Kepler conjecture , 1998, math/9811078.

[31]  Noam D. Elkies,et al.  On the packing densities of superballs and other bodies , 1991 .

[32]  James E. Humphreys,et al.  Reflection groups and Coxeter groups: Coxeter groups , 1990 .

[33]  N. J. A. Sloane,et al.  An improvement to the Minkowski-Hiawka bound for packing superballs , 1987 .

[34]  C. Dunkl.  Cube group invariant spherical harmonics and Krawtchouk polynomials , 1981 .

[35]  Richard P. Stanley,et al.  Invariants of finite groups and their applications to combinatorics , 1979 .

[36]  E. Stein,et al.  Introduction to Fourier Analysis on Euclidean Spaces. , 1971 .

[37]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.

[38]  Helmut Groemer.  Über die dichteste gitterförmige Lagerung kongruenter Tetraeder , 1962 .

[39]  Sabrina Hirsch,et al.  Reflection Groups And Coxeter Groups , 2016 .

[40]  Günter M. Ziegler,et al.  Three Mathematics Competitions , 2011 .

[41]  S. Torquato,et al.  Dense packings of the Platonic and Archimedean solids , 2009 .

[42]  Randall R. Holmes.  Linear Representations of Finite Groups , 2008 .

[43]  M. Kojima,et al.  SDPA-C (SemiDefinite Programming Algorithm - Completion method) User's Manual — Version 6.2.0 , 2004 .

[44]  D. Kamenetsky.  Symmetry Groups , 2003 .

[45]  John P. D'Angelo,et al.  Inequalities from Complex Analysis , 2002 .

[46]  A. Terras.  Fourier Analysis on Finite Groups and Applications: Index , 1999 .

[47]  Olga Taussky-Todd.  SOME CONCRETE ASPECTS OF HILBERT'S 17TH PROBLEM , 1996 .

[48]  Bernd Sturmfels,et al.  Algorithms in invariant theory , 1993, Texts and monographs in symbolic computation.

[49]  J. J. Seidel,et al.  Cubature Formulae, Polytopes, and Spherical Designs , 1981 .

[50]  Douglas J. Hoylman.  THE DENSEST LATTICE PACKING OF TETRAHEDRA , 1970 .

[51]  H. Minkowski.  Dichteste gitterförmige Lagerung kongruenter Körper , 1904 .

[52]  Ludwig August Seeber.  Recension der "Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen von Ludwig August Seeber". , 1840 .