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Intersection bodies and the Busemann-Petty problem


Author: R. J. Gardner
Journal: Trans. Amer. Math. Soc. 342 (1994), 435-445
MSC: Primary 52A38; Secondary 52A40
DOI: https://doi.org/10.1090/S0002-9947-1994-1201126-7
MathSciNet review: 1201126
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Abstract: It is proved that the answer to the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies in d-dimensional Euclidean space $ {\mathbb{E}^d}$ is negative for a given d if and only if certain centrally symmetric convex bodies exist in $ {\mathbb{E}^d}$ which are not intersection bodies. It is also shown that a cylinder in $ {\mathbb{E}^d}$ is an intersection body if and only if $ d \leq 4$, and that suitably smooth axis-convex bodies of revolution are intersection bodies when $ d \leq 4$. These results show that the Busemann-Petty problem has a negative answer for $ d \geq 5$ and a positive answer for $ d = 3$ and $ d = 4$ when the body with smaller sections is a body of revolution.


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  • [B] K. Ball, Some remarks on the geometry of convex sets, Geometric Aspects of Functional Analysis, (J. Lindenstrauss and V. D. Milman, eds.), Lecture Notes in Math., vol. 1317, Springer-Verlag, Berlin and New York, 1988, pp. 224-231. MR 950983 (89h:52009)
  • [Be] M. Berger, Convexity, Amer. Math. Monthly 97 (1990), 650-678. MR 1072810 (91f:52001)
  • [Bo] J. Bourgain, On the Busemann-Petty problem for perturbations of the ball, Geom. Funct. Anal. 1 (1991), 1-13. MR 1091609 (92c:52008)
  • [BL] J. Bourgain and J. Lindenstrauss, Projection bodies, Geometric Aspects of Functional Analysis, (J. Lindenstrauss and V. D. Milman, eds.), Lecture Notes in Math., vol. 1317, Springer-Verlag, Berlin and New York, 1988, pp. 250-270. MR 950986 (89g:46024)
  • [BZ] Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Springer-Verlag, Berlin and New York, 1988. MR 936419 (89b:52020)
  • [B$ _{1}$] H. Busemann, Volume in terms of concurrent cross-sections, Pacific J. Math. 3 (1953), 1-12. MR 0055712 (14:1115e)
  • [B$ _{2}$] -, Volumes and areas of cross-sections, Amer. Math. Monthly 67 (1960), 248-250; correction 67 (1960), 671. MR 0120562 (22:11313)
  • [BP] H. Busemann and C. M. Petty, Problems on convex bodies, Math. Scand. 4 (1956), 88-94. MR 0084791 (18:922b)
  • [CFG] H. T. Croft, K. J. Falconer, and R. K. Guy, Unsolved problems in geometry, Springer-Verlag, Berlin and New York, 1991. MR 1107516 (92c:52001)
  • [F] P. Funk, Über Flächen mit lauter geschlossenen geodätischen Linien, Math. Ann. 74 (1913), 278-300. MR 1511763
  • [Ga] R. J. Gardner, A positive answer to the Busemann-Petty problem in three dimensions (to appear). MR 1298719 (95i:52005)
  • [G] A. A. Giannopoulos, A note on a problem of H. Busemann and C. M. Petty concerning sections of symmetric convex bodies, Mathematika 37 (1990), 239-244. MR 1099772 (92c:52009)
  • [Gi] M. Giertz, A note on a problem of Busemann, Math. Scand. 25 (1969), 145-148. MR 0262929 (41:7534)
  • [GLW] P. R. Goodey, E. Lutwak and W. Weil, Functional analytic characterizations of classes of convex bodies (to appear).
  • [Gr] E. L. Grinberg, Spherical harmonics and integral geometry on projective spaces, Trans. Amer. Math. Soc. 279 (1983), 187-203. MR 704609 (84m:53071)
  • [GR] E. L. Grinberg and I. Rivin, Infinitesimal aspects of the Busemann-Petty problem, Bull. London Math. Soc. 22 (1990), 478-484. MR 1082020 (92e:52012)
  • [H] H. Hadwiger, Radialpotenzintegrale zentralsymmetrischer Rotationskörper und ungleichheitaussagen Busemannischer Art, Math. Scand. 23 (1968), 193-200. MR 0254739 (40:7946)
  • [He$ _{1}$] S. Helgason, Groups and geometric analysis, Academic Press, San Diego, 1984. MR 754767 (86c:22017)
  • [He$ _{2}$] -, The totally-geodesic Radon transform on constant curvature spaces, Contemp. Math. 113 (1990), 141-149. MR 1108651 (92j:53036)
  • [K] V. L. Klee, Ungelöstes Problem Nr. 44, Elem. Math. 17 (1962), 84.
  • [KN] S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol. 2, Wiley, New York, 1969.
  • [LR] D. G. Larman and C. A. Rogers, The existence of a centrally symmetric convex body with central sections that are unexpectedly small, Mathematika 22 (1975), 164-175. MR 0390914 (52:11737)
  • [L$ _{1}$] E. Lutwak, Intersection bodies and dual mixed volumes, Adv. Math. 71 (1988), 232-261. MR 963487 (90a:52023)
  • [L$ _{2}$] -, On some ellipsoid formulas of Busemann, Furstenberg and Tzkoni, Guggenheimer, and Petty, J. Math. Anal. Appl. 159 (1991), 18-26. MR 1119418 (92k:52014)
  • [MP] V. D. Milman and A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, Geometric Aspects of Functional Analysis, (J. Lindenstrauss and V. D. Milman, eds.), Lecture Notes in Math., vol. 1376, Springer-Verlag, Berlin and New York, 1989, pp. 64-104. MR 1008717 (90g:52003)
  • [O] V. I. Oliker, Hypersurfaces in $ {\mathbb{R}^{n + 1}}$ with prescribed Gaussian curvature and related equations of Monge-Ampére type, Comm. Partial Differential Equations 9 (1984), 807-838. MR 748368 (85h:53047)
  • [P] M. Papadimitrakis, On the Busemann-Petty problem about convex, centrally symmetric bodies in $ {\mathbb{R}^n}$, Mathematika (to appear). MR 1203282 (94a:52019)
  • [Pe] C. M. Petty, Centroid surfaces, Pacific J. Math. 11 (1961), 1535-1547. MR 0133733 (24:A3558)
  • [S$ _{1}$] R. Schneider, Functions on a sphere with vanishing integrals over certain subspheres, J. Math. Anal. Appl. 26 (1969), 381-384. MR 0237723 (38:6004)
  • [S$ _{2}$] -, Smooth approximations of convex bodies, Rend. Circ. Mat. Palermo (2) 33 (1984), 436-440. MR 779946 (86e:52004)
  • [SW] R. Schneider and W. Weil, Convexity and Its Applications, (P. M. Gruber and J. M. Wills, eds.), Birkhäuser, Basel, 1983, pp. 296-317. MR 731116 (85c:52010)
  • [T] S. Tanno, Central sections of centrally symmetric convex bodies, Kodai Math. J. 10 (1987), 343-361. MR 929994 (90c:52009)
  • [Z$ _{1}$] Gaoyong Zhang, Centered bodies and dual mixed volumes (to appear).
  • [Z$ _{2}$] -, Intersection bodies and the four-dimensional Busemann-Petty problem, Duke Math. J. 71 (1993), 233-240. MR 1230300 (94f:52007)
  • [Z$ _{3}$] -, Intersection bodies and the Busemann-Petty inequalities in $ {\mathbb{R}^n}$, Ann. of Math, (to appear).

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1201126-7
Keywords: Convex body, section, Busemann-Petty problem, intersection body
Article copyright: © Copyright 1994 American Mathematical Society

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