Download Discrete Geometry and Optimization by Javier Alonso, Horst Martini, Margarita Spirova (auth.), PDF
By Javier Alonso, Horst Martini, Margarita Spirova (auth.), Karoly Bezdek, Antoine Deza, Yinyu Ye (eds.)
Optimization has lengthy been a resource of either concept and purposes for geometers, and conversely, discrete and convex geometry have supplied the principles for lots of optimization innovations, resulting in a wealthy interaction among those topics. the aim of the Workshop on Discrete Geometry, the convention on Discrete Geometry and Optimization, and the Workshop on Optimization, held in September 2011 on the Fields Institute, Toronto, used to be to additional stimulate the interplay among geometers and optimizers. This quantity displays the interaction among those components.
The inspiring Fejes Tóth Lecture sequence, added by means of Thomas Hales of the college of Pittsburgh, exemplified this technique. whereas those fields have lately witnessed loads of job and successes, many questions stay open. for instance, Fields medalist Stephen Smale acknowledged that the query of the lifestyles of a strongly polynomial time set of rules for linear optimization is among the most crucial unsolved difficulties first and foremost of the twenty first century. The large diversity of subject matters lined during this quantity demonstrates the various fresh and fruitful connections among diversified methods, and lines novel effects and cutting-edge surveys in addition to open difficulties.
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Extra resources for Discrete Geometry and Optimization
Example text
0:6/ D 2:2 for j D 1; 3; 5. 2:2/ D 13:6; i iD1 which implies (5). With an upper bound for Nmax determined, a lower bound for the left-hand side of (6) can be obtained using the Delsarte bound for spherical codes. Specifically, 2 3 C D fxi gm 0:6843. 2RD / the usual distance distribution of the code to be the function ˛. / W Œ 1; 1 ! s/ D It is then easy to see that ˛. s/ D m 1: (8) 1ÄsÄz Let ˚k . s;s/ is a Jacobi polynomial. t / where Pk Chaps. s/ 0; k D 1; 2; : : : : (9) 1ÄsÄz From (8) and (9), using k D 1; : : : ; d , a bound on the left-hand side of (6) can be obtained via the semi-infinite linear programming problem An Approach to the Dodecahedral Conjecture Based on Bounds for Spherical Codes 41 Fig.
3 Applying Bounds for Spherical Codes We now describe an approach to proving the key inequality (1) based on bounds for spherical codes. A set C D fxi gm <3 is called a spherical z-code if kxi k D 1 iD1 T for each i , and xi xj Ä z for all i ¤ j . For example, a packing of unit spheres that all touch (or “kiss”) a unit sphere centered at the origin generates a spherical 1=2-code. To begin we establish that if R is sufficiently small and fxN i gm iD1 are points with 1 Ä kxN i k Ä R for each i and kxN i xN j k 1 for all i ¤ j , then the normalized points xi D x=k N xN i k form a z-code for a suitable value of z.
In recent work, Hales [7] announced a proof of the “strong” dodecahedral conjecture, which is the original dodecahedral conjecture with surface area replacing volume throughout. The proof methodology of [7] also utilizes Fejes T´oth’s key inequality, which is apparently the basis for a new computational proof of the Kepler conjecture in [8]. These recent developments suggest that continued efforts to provide a direct proof of the key inequality remain a very interesting topic for further research.