By Virgil Snyder, Charles Herschel Sisam

ISBN-10: 1178217175

ISBN-13: 9781178217179

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**Extra resources for Analytic Geometry of Space**

**Sample text**

L EMMA 2. If the Ekeland–Hofer capacities and the volume capacity form a generating system for symplectic capacities on Ell2n , then c B 41 D 12 . 2 C "/ for every " > 0. P ROOF OF L EMMA 2. We can assume that all capacities are normalized. By assumption, there exists a sequence fi of homogeneous and monotone functions in the cNk and in cvol forming normalized capacities which pointwise converge to QUANTITATIVE SYMPLECTIC GEOMETRY 31 c B . As is easy to see, cNk E 14 ; 1 Ä cNk B 4 12 for all k, and cvol E 14 ; 1 D cvol B 4 12 .

Ginzburg and E. Kerman, Periodic orbits in magnetic fields in dimensions greater than two, Geometry and topology in dynamics (Winston-Salem, NC, 1998, and San Antonio, TX, 1999), 113–121, Contemp. Math. 246, American Mathematical Society, Providence, RI, 1999. [38] E. DG/0311460. [39] M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82, 307–347 (1985). [40] D. Hermann, Holomorphic curves and Hamiltonian systems in an open set with restricted contact-type boundary, Duke Math.

Eur. Math. Soc. 1, 87–107 (1999). [82] L. Polterovich, The geometry of the group of symplectic diffeomorphisms, Lectures in Mathematics ETH Z¨urich, Birkh¨auser, Basel, 2001. [83] P. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31, 157–184 (1978). [84] P. Rabinowitz, Periodic solutions of a Hamiltonian system on a prescribed energy surface, J. Differential Equations 33, 336–352 (1979). [85] F. Schlenk, Symplectic embedding of ellipsoids, Israel J. of Math. 138, 215–252 (2003).

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