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Polar duals of convex bodies


Author: Mostafa Ghandehari
Journal: Proc. Amer. Math. Soc. 113 (1991), 799-808
MSC: Primary 52A39
MathSciNet review: 1057954
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Abstract: A generalization and the dual version of the following result due to Firey is given: The mixed area of a plane convex body and its polar dual is at least $ \pi $. We give a sharp upper bound for the product of the dual cross-sectional measure of any index and that of its polar dual. A general result for a convex body $ K$ and a convex increasing real-valued function gives inequalities for sets of constant width and sets with equichordal points as special cases.


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  • [1] R. P. Bambah, Polar reciprocal convex bodies, Proc. Cambridge Philos. Soc. 51 (1955), 377–378. MR 0070201
  • [2] T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, reprint, Chelsea, New York, 1948.
  • [3] Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ; Springer Series in Soviet Mathematics. MR 936419
  • [4] G. D. Chakerian, Geometric inequalities for plane convex bodies, Canad. Math. Bull. 22 (1979), no. 1, 9–16. MR 532264, 10.4153/CMB-1979-002-3
  • [5] G. D. Chakerian, Mixed areas and the self-circumference of a plane convex body, Arch. Math. (Basel) 34 (1980), no. 1, 81–83. MR 575554, 10.1007/BF01224933
  • [6] G. D. Chakerian and H. Groemer, Convex bodies of constant width, Convexity and its applications, Birkhäuser, Basel, 1983, pp. 49–96. MR 731106
  • [7] Alexander Dinghas, Minkowskische Summen und Integrale. Superadditive Mengenfunktionale. Isoperimetrische Ungleichungen, Mémor. Sci. Math., Fasc. 149, Gauthier-Villars, Paris, 1961 (German). MR 0132456
  • [8] A. Dvoretzky and C. A. Rogers, Absolute and unconditional convergence in normed linear spaces, Proc. Nat. Acad. Sci. U. S. A. 36 (1950), 192–197. MR 0033975
  • [9] H. G. Eggleston, Convexity, Cambridge Tracts in Mathematics and Mathematical Physics, No. 47, Cambridge University Press, New York, 1958. MR 0124813
  • [10] Wm. J. Firey, The mixed area of a convex body and its polar reciprocal, Israel J. Math. 1 (1963), 201–202. MR 0163213
  • [11] Wm. J. Firey, Support flats to convex bodies, Geometriae Dedicata 2 (1973), 225–248. MR 0397548
  • [12] H. Guggenheimer, The analytic geometry of the unsymmetric Minkowski plane, Lecture Notes, University of Minnesota, Minneapolis, 1967.
  • [13] -, The analytic geometry of the Minkowski plane. I, A universal isoperimetric inequality, Abstract 642-697, Notices Amer. Math. Soc. 14 (1967), 121.
  • [14] H. Guggenheimer, Hill equations with coexisting periodic solutions, J. Differential Equations 5 (1969), 159–166. MR 0239193
  • [15] H. Guggenheimer, Polar reciprocal convex bodies, Israel J. Math. 14 (1973), 309–316. MR 0320888
  • [16] H. Guggenheimer, Correction to: “Polar reciprocal convex bodies” (Israel J. Math. 14 (1973), no. 3, 309–316), Israel J. Math. 29 (1978), no. 2-3, 312. MR 0467527
  • [17] H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957 (German). MR 0102775
  • [18] Erhard Heil, Ungleichungen für die Quermassintegrale polarer Körper, Manuscripta Math. 19 (1976), no. 2, 143–149. MR 0405241
  • [19] C. G. Lekkerkerker, Geometry of numbers, Bibliotheca Mathematica, Vol. VIII, Wolters-Noordhoff Publishing, Groningen; North-Holland Publishing Co., Amsterdam-London, 1969. MR 0271032
  • [20] Erwin Lutwak, On cross-sectional measures of polar reciprocal convex bodies, Geometriae Dedicata 5 (1976), no. 1, 79–80. MR 0428197
  • [21] Erwin Lutwak, On the Blaschke-Santaló inequality, Discrete geometry and convexity (New York, 1982) Ann. New York Acad. Sci., vol. 440, New York Acad. Sci., New York, 1985, pp. 106–112. MR 809197, 10.1111/j.1749-6632.1985.tb14544.x
  • [22] Erwin Lutwak, Intersection bodies and dual mixed volumes, Adv. in Math. 71 (1988), no. 2, 232–261. MR 963487, 10.1016/0001-8708(88)90077-1
  • [23] V. Klee, Shapes of the future--some unsolved problems in geometry, Part I, Two dimensions, Film available MAA, 1971.
  • [24] F. Steinhardt, On distance functions and on polar series of convex bodies, Ph.D. Thesis, Columbia Univ., 1951.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1057954-7
Keywords: Convex body, polar dual, mixed volume, dual mixed volume
Article copyright: © Copyright 1991 American Mathematical Society