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PARALLEL SECTIONS HOMOTHETY BODIES WITH MINIMAL MAHLER VOLUME IN ℝn

  • Lin, Youjiang ;
  • Leng, Gangsong
  • Received : 2013.08.06
  • Published : 2014.07.31

Abstract

In the paper, we define a class of convex bodies in $\mathbb{R}^n$-parallel sections homothety bodies, and for some special parallel sections homothety bodies, we prove that n-cubes have the minimal Mahler volume.

Keywords

convex body;polar body;parallel sections homothety bodies;Mahler conjecture;cylinder

References

  1. S. Reisner, Minimal volume product in Banach spaces with a 1-unconditional basis, J. London Math. Soc. 36 (1987), no.1, 126-136.
  2. J. Saint-Raymond, Sur le volume des corps convexes sym etriques, Seminaire d'initiation 'al' Analyse, 1980/1981, Publ. Math. Univ. Pierre et Marie Curie, Paris, 1981, 1-25.
  3. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia Math. Appl., vol. 44, Cambridge University Press, Cambridge, 1993.
  4. T. Tao, Structure and Randomness, Pages from year one of a mathematical blog. American Mathematical Society, Providence, RI, 2008.
  5. Y. Gordon, M. Meyer, and S. Reisner, Zonoids with minimal volume-product-a new proof, Proc. Amer. Math. Soc. 104 (1988), no. 1, 273-276.
  6. J. Kim and S. Reisner, Local minimality of the volume-product at the simplex, Mathe-matika, in press.
  7. G. Kuperberg, From the Mahler conjecture to Gauss linking integrals, Geom. Funct. Anal. 18 (2008), no. 3, 870-892. https://doi.org/10.1007/s00039-008-0669-4
  8. Y. Lin and G. Leng, Convex bodies with minimal volume product in $\mathbb{R}^2$-a new proof, Discrete Math. 310 (2010), no. 21, 3018-3025. https://doi.org/10.1016/j.disc.2010.07.008
  9. M. A. Lopez and S. Reisner, A special case of Mahler's conjecture, Discrete Comput. Geom. 20 (1998), no. 2, 163-177. https://doi.org/10.1007/PL00000076
  10. E. Lutwak, D. Yang, and G. Zhang, A volume inequality for polar bodies, J. Differential Geom. 84 (2010), no. 1, 163-178. https://doi.org/10.4310/jdg/1271271796
  11. K. Mahler, Ein Ubertragungsprinzip fur konvexe Korper, Casopis Pest. Mat. Fys. 68 (1939), 93-102.
  12. K. Mahler, Ein Minimalproblem fur konvexe Polygone, Mathematica (Zutphen) B. 7 (1939), 118-127.
  13. M. Meyer, Une caracterisation volumique de certains espaces normes de dimension finie, Israel J. Math. 55 (1986), no. 3, 317-326. https://doi.org/10.1007/BF02765029
  14. M. Meyer, Convex bodies with minimal volume product in $\mathbb{R}^2$, Monatsh. Math. 112 (1991), no. 4, 297-301. https://doi.org/10.1007/BF01351770
  15. M. Meyer and S. Reisner, Shadow systems and volumes of polar convex bodies, Mathe-matika 53 (2006), no. 1, 129-148.
  16. F. Nazarov, F. Petrov, D. Ryabogin, and A. Zvavitch, A remark on the Mahler conjecture: local minimality of the unit cube, Duke Math. J. 154 (2010), no. 3, 419-430. https://doi.org/10.1215/00127094-2010-042
  17. S. Reisner, Random polytopes and the volume-product of symmetric convex bodies, Math. Scand. 57 (1985), no. 2, 386-392. https://doi.org/10.7146/math.scand.a-12124
  18. S. Reisner, Zonoids with minimal volume-product, Math. Z. 192 (1986), no. 3, 339-346. https://doi.org/10.1007/BF01164009
  19. S. Artstein, B. Klartag, and V. D. Milman, On the Santalo point of a function and a functional Santalo inequality, Mathematika 54 (2004), 33-48.
  20. K. Ball, Mahlers conjecture and wavelets, Discrete Comput. Geom. 13 (1995), no. 3-4, 271-277. https://doi.org/10.1007/BF02574044
  21. J. Bourgain and V. D. Milman, New volume ratio properties for convex symmetric bodies in $\mathbb{R}^n$, Invent. Math. 88 (1987), no. 2, 319-340. https://doi.org/10.1007/BF01388911
  22. S. Campi and P. Gronchi, Volume inequalities for $L_p$-zonotopes, Mathematika 53 (2006), no. 1, 71-80. https://doi.org/10.1112/S0025579300000036
  23. S. Campi and P. Gronchi, On volume product inequalities for convex sets, Proc. Amer. Math. Soc. 134 (2006), no. 8, 2393-2402. https://doi.org/10.1090/S0002-9939-06-08241-4
  24. S. Campi and P. Gronchi, Extremal convex sets for Sylvester-Busemann type functionals, Appl. Anal. 85 (2006), no. 1-3, 129-141.
  25. M. Fradelizi, Y. Gordon, M. Meyer, and S. Reisner, The case of equality for an inverse Santalo functional inequality, Adv. Geom. 10 (2010), no. 4, 621-630.
  26. R. J. Gardner, Geometric Tomography, Second edition. Encyclopedia of Mathematics and its Applications, 58. Cambridge University Press, Cambridge, 2006.