A Generalized Affine Isoperimetric Inequality

Preprint English OPEN
Chen, Wenxiong ; Howard, Ralph ; Lutwak, Erwin ; Yang, Deane ; Zhang, Gaoyong (2004)
  • Subject: 52A40 (Primary) 52A10 53A15 (Secondary) | Mathematics - Classical Analysis and ODEs | Mathematics - Metric Geometry
    arxiv: Mathematics::Metric Geometry | Mathematics::Differential Geometry
A purely analytic proof is given for an inequality that has as a direct consequence the two most important affine isoperimetric inequalities of plane convex geometry: The Blaschke-Santalo inequality and the affine isoperimetric inequality of affine differential geometry.
  • References (17)
    17 references, page 1 of 2

    [1] G. Dolzmann and D. Hug, Equality of two representations of extended affine surface area, Arch. Math. (Basel) 65 (1995), no. 4, 352-356. MR 97c:52019

    [2] G. H. Hardy, J. E. Littlewood, and G. P´olya, Inequalities, Cambridge, at the University Press, 1952, 2d ed. MR 13,727e

    [3] Evans M. Harrell, II, A direct proof of a theorem of Blaschke and Lebesgue, J. Geom. Anal. 12 (2002), no. 1, 81-88. MR 1 881 292

    [4] E. Hewitt and K. Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, Springer-Verlag, New York, 1965. MR 32 #5826

    [5] L. H¨ormander, The analysis of linear partial differential operators. I, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1990, Distribution theory and Fourier analysis.

    [6] Kurt Leichtweiß, Zur Affinoberfl¨ache konvexer K¨orper, Manuscripta Math. 56 (1986), no. 4, 429-464. MR 87k:52011

    [7] E. Lutwak, On the Blaschke-Santal´o inequality, Discrete geometry and convexity (New York, 1982), New York Acad. Sci., New York, 1985, pp. 106-112. MR 87c:52018

    [8] , Extended affine surface area, Adv. Math. 85 (1991), no. 1, 39-68. MR 92d:52012

    [9] E. Lutwak, The Brunn-Minkowski-Firey theory I: mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), 131-150.

    [10] E. Lutwak, V. Oliker, On the regularity of solutions to a generalization of the Minkowski problem, J. Differential Geom. 41 (1995), 227-246.

  • Metrics
    No metrics available
Share - Bookmark