Convex Bodies With Minimal Volume Product in R^2 --- A New Proof

Preprint English OPEN
Lin, Youjiang (2010)
  • Subject: Mathematics - Metric Geometry | 52A10, 52A40

In this paper, a new proof of the following result is given: The product of the volumes of an origin symmetric convex bodies $K$ in R^2 and of its polar body is minimal if and only if $K$ is a parallelogram.
  • References (14)
    14 references, page 1 of 2

    [1] J. Bourgain, V. D. Milman, New volume ratio properties for convex symmetric bodies in Rn. Invent. Math. 88 (1987),319-340. MR0880954 (88f:52013)

    [2] Y. Gordon, M. Meyer and S. Reisner, Zonoids with minimal volumeCproduct - a new proof. Proceedings of the American Math. Soc. 104 (1988), 273-276. MR0958082 (89i:52015)

    [3] G. Kuperberg, From the Mahler Conjecture to Gauss Linking Integrals. Geometric And Functional Analysis, 18 (2008), 870-892. MR2438998 (2009i:52005)

    [4] K. Mahler, Ein Ubertragungsprinzip fur konvexe Korper. Casopis Pyest. Mat. Fys. 68, (1939), 93-102. MR0001242 (1,202c)

    [5] E. Lutwak, G. Zhang, Blaschke-Santalo┬┤ inequalities. J. Differential Geom. 47 (1997), 1-16. 52A40 MR1601426 (2000c:52011)

    [6] M. Meyer, Une caracterisation volumique de certains espaces normes de dimension finie. Israel J. Math. 55 (1986), 317-326. MR0876398 (88f:52017)

    [7] M. Meyer, Convex bodies with minimal volume product in R2. Monatsh. Math. 112 (1991), 297-301. MR1141097 (92k:52015)

    [8] M. Meyer and A. Pajor, On Santalo┬┤ inequality. Geometric aspects of functional analysis (1987-88), Lecture Notes in Math., 1376, Springer, Berlin, (1989), 261-263. MR1008727 (90h:52012)

    [9] C. M. Petty, Affine isoperimetric problems. Discrete geometry and convexity (New York, 1982), 113-127, Ann. New York Acad. Sci., 440, New York Acad. Sci., New York, 1985. MR0809198 (87a:52014)

    [10] S. Reisner, Zonoids with minimal volume-product. Math. Zeitschrift 192 (1986), 339-346. MR0958082 (89i:52015)

  • Metrics
    No metrics available
Share - Bookmark