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A proof of the Kepler conjecture
 Math. Intelligencer
, 1994
"... This section describes the structure of the proof of ..."
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Cited by 112 (11 self)
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This section describes the structure of the proof of
Computer Assisted Proof of Optimal Approximability Results
, 2002
"... We obtain computer assisted proofs of several spherical volume inequalities that appear in the analysis of semidefinite programming based approximation algorithms for Boolean constraint satisfaction problems. These inequalities imply, in particular, that the performance ratio achieved by the MAX 3S ..."
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Cited by 13 (4 self)
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We obtain computer assisted proofs of several spherical volume inequalities that appear in the analysis of semidefinite programming based approximation algorithms for Boolean constraint satisfaction problems. These inequalities imply, in particular, that the performance ratio achieved by the MAX 3SAT approximation algorithm of Karloff and Zwick is indeed 7/8, as conjectured by them, and that the performance ratio of the MAX 3CSP algorithm of the author is indeed ½. Other results are also implied. The computer assisted proofs are obtained using a system called REALSEARCH written by the author. This system uses interval arithmetic to produce rigorous proofs that certain collections of constraints in real variables have no real solution.
Sphere packings in 3space
, 2004
"... In this paper we survey results on packings of congruent spheres in 3dimensional spaces of constant curvature. The topics discussed are as follows: Hadwiger numbers of convex bodies and kissing numbers of spheres; Touching numbers of convex bodies; Newton numbers of convex bodies; Onesided Had ..."
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Cited by 5 (0 self)
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In this paper we survey results on packings of congruent spheres in 3dimensional spaces of constant curvature. The topics discussed are as follows: Hadwiger numbers of convex bodies and kissing numbers of spheres; Touching numbers of convex bodies; Newton numbers of convex bodies; Onesided Hadwiger and kissing numbers; Contact graphs of finite packings and the combinatorial Kepler problem; Isoperimetric problems for Voronoi cells and the strong dodecahedral conjecture; The strong Kepler conjecture. Each topic is discussed in details along with some open problems. Four topics from the above list are treated in spaces of dimension different from three as well. 1