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A Fourier analytic proof of the Blaschke-Santaló Inequality


Authors: Gabriele Bianchi and Michael Kelly
Journal: Proc. Amer. Math. Soc. 143 (2015), 4901-4912
MSC (2010): Primary 52A40, 42A05, 46E22
DOI: https://doi.org/10.1090/proc/12785
Published electronically: July 10, 2015
MathSciNet review: 3391048
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Abstract: The Blaschke-Santaló Inequality is the assertion that the volume product of a centrally symmetric convex body in Euclidean space is maximized by (and only by) ellipsoids. In this paper we give a Fourier analytic proof of this fact.


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  • [Ahi48] N. I. Ahiezer, On the theory of entire functions of finite degree, Doklady Akad. Nauk SSSR (N.S.) 63 (1948), 475-478 (Russian). MR 0027333 (10,289h)
  • [AB02] V. V. Arestov and E. E. Berdysheva, The Turán problem for a class of polytopes, East J. Approx. 8 (2002), no. 3, 381-388. MR 1932340 (2003i:42010)
  • [Bia13] Gabriele Bianchi, The covariogram and Fourier-Laplace transform in $ \mathbb{C}^n$, 2013, arXiv:1312.7816 [math.MG].
  • [Bla17] W. Blaschke, Über affine Geometrie VII: Neue Extremeingenschaften von Ellipse und Ellipsoid, Ber. Verh. Sächs. Akad. Wiss., Math. Phys. Kl. 69 (1917), 412-420.
  • [BM87] J. Bourgain and V. D. Milman, New volume ratio properties for convex symmetric bodies in $ {\bf R}^n$, Invent. Math. 88 (1987), no. 2, 319-340. MR 880954 (88f:52013), https://doi.org/10.1007/BF01388911
  • [Boa54] Ralph Philip Boas Jr., Entire functions, Academic Press Inc., New York, 1954. MR 0068627 (16,914f)
  • [Cas92] J. W. S. Cassels, Obituary: Kurt Mahler, Bull. London Math. Soc. 24 (1992), no. 4, 381-397. MR 1165384 (93f:01016), https://doi.org/10.1112/blms/24.4.381
  • [dB68] Louis de Branges, Hilbert spaces of entire functions, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1968. MR 0229011 (37 #4590)
  • [EGR04] Werner Ehm, Tilmann Gneiting, and Donald Richards, Convolution roots of radial positive definite functions with compact support, Trans. Amer. Math. Soc. 356 (2004), no. 11, 4655-4685 (electronic). MR 2067138 (2005g:42012), https://doi.org/10.1090/S0002-9947-04-03502-0
  • [Gor01] D. V. Gorbachev, An extremal problem for periodic functions with support in a ball, Mat. Zametki 69 (2001), no. 3, 346-352 (Russian, with Russian summary); English transl., Math. Notes 69 (2001), no. 3-4, 313-319. MR 1846833 (2002e:42006), https://doi.org/10.1023/A:1010275206760
  • [HV96] Jeffrey J. Holt and Jeffrey D. Vaaler, The Beurling-Selberg extremal functions for a ball in Euclidean space, Duke Math. J. 83 (1996), no. 1, 202-248. MR 1388849 (97f:30038), https://doi.org/10.1215/S0012-7094-96-08309-X
  • [Kim13] Jaegil Kim, Minimal volume product near Hanner polytopes, J. Funct. Anal. 266 (2014), no. 4, 2360-2402. MR 3150164, https://doi.org/10.1016/j.jfa.2013.08.008
  • [KR06] Mihail N. Kolountzakis and Szilárd Gy. Révész, Turán's extremal problem for positive definite functions on groups, J. London Math. Soc. (2) 74 (2006), no. 2, 475-496. MR 2269590 (2007k:43009), https://doi.org/10.1112/S0024610706023234
  • [Mah39] Kurt Mahler, Ein Minimalproblem für Konvexe Polygone, Mathematica (Zutphen) B (1939), 118-127.
  • [MP90] Mathieu Meyer and Alain Pajor, On the Blaschke-Santaló inequality, Arch. Math. (Basel) 55 (1990), no. 1, 82-93. MR 1059519 (92b:52013), https://doi.org/10.1007/BF01199119
  • [MR89] M. Meyer and S. Reisner, Characterizations of ellipsoids by section-centroid location, Geom. Dedicata 31 (1989), no. 3, 345-355. MR 1025195 (90m:52006), https://doi.org/10.1007/BF00147465
  • [Naz12] Fedor Nazarov, The Hörmander proof of the Bourgain-Milman theorem, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 2050, Springer, Heidelberg, 2012, pp. 335-343. MR 2985302, https://doi.org/10.1007/978-3-642-29849-3_20
  • [NPRZ10] Fedor Nazarov, Fedor Petrov, Dmitry Ryabogin, and Artem Zvavitch, A remark on the Mahler conjecture: local minimality of the unit cube, Duke Math. J. 154 (2010), no. 3, 419-430. MR 2730574 (2012a:52010), https://doi.org/10.1215/00127094-2010-042
  • [Pet85] C. M. Petty, Affine isoperimetric problems, Discrete geometry and convexity (New York, 1982) Ann. New York Acad. Sci., vol. 440, New York Acad. Sci., New York, 1985, pp. 113-127. MR 809198 (87a:52014), https://doi.org/10.1111/j.1749-6632.1985.tb14545.x
  • [PP37] M. Plancherel and G. Pólya, Fonctions entières et intégrales de fourier multiples, Comment. Math. Helv. 10 (1937), no. 1, 110-163 (French). MR 1509570, https://doi.org/10.1007/BF01214286
  • [Rei85] Shlomo Reisner, Random polytopes and the volume-product of symmetric convex bodies, Math. Scand. 57 (1985), no. 2, 386-392. MR 832364 (87g:52011)
  • [Rei86] Shlomo Reisner, Zonoids with minimal volume-product, Math. Z. 192 (1986), no. 3, 339-346. MR 845207 (87g:52022), https://doi.org/10.1007/BF01164009
  • [RSN55] Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727 (17,175i)
  • [RZ] Dmitry Ryabogin and Artem Zvavitch, Analytic methods in convex geometry, in preparation.
  • [Sai81] J. Saint-Raymond, Sur le volume des corps convexes symétriques, Initiation Seminar on Analysis: G. Choquet-M. Rogalski-J. Saint-Raymond, 20th Year: 1980/1981, Publ. Math. Univ. Pierre et Marie Curie, vol. 46, Univ. Paris VI, Paris, 1981, pp. Exp. No. 11, 25 (French). MR 670798 (84j:46033)
  • [San49] L. A. Santaló, Un invariante afin para los cuerpos convexos del espacio des n dimensiones, Portugaliae Math. 8 (1949), 155-161. MR 0039293 (12,526f)
  • [Sch14] Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Second expanded edition, Encyclopedia of Mathematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014. MR 3155183
  • [Sie35] Carl Ludwig Siegel, Über Gitterpunkte in Convexen Körpern und ein Damit Zusammenhängendes Extremalproblem, Acta Math. 65 (1935), no. 1, 307-323 (German). MR 1555407, https://doi.org/10.1007/BF02420949
  • [SW71] Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972 (46 #4102)
  • [Tao08] Terence Tao, Structure and randomness, American Mathematical Society, Providence, RI, 2008. Pages from year one of a mathematical blog. MR 2459552 (2010h:00002)

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Additional Information

Gabriele Bianchi
Affiliation: Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze
Email: gabriele.bianchi@unifi.it

Michael Kelly
Affiliation: Department of Mathematics, University of Texas
Email: mkelly@math.utexas.edu

DOI: https://doi.org/10.1090/proc/12785
Received by editor(s): February 10, 2014
Received by editor(s) in revised form: August 2, 2014
Published electronically: July 10, 2015
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2015 American Mathematical Society

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