Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format
Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(e) ISSN 0273-0979(p)

     

On the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies

Author(s): R. J. Gardner
Journal: Bull. Amer. Math. Soc. 30 (1994), 222-226.
MSC (2000): Primary 52A38; Secondary 52A40
MathSciNet review: 1246466
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We present a method which shows that in $                 {\mathbb{E}^3}$ the Busemann-Petty problem, concerning central sections of centrally symmetric convex bodies, has a positive answer. Together with other results, this settles the problem in each dimension.


References:

[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, 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, 1988, pp. 250-270. MR 950986 (89g:46024)

[BZ]
Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Springer-Verlag, Berlin, 1988. MR 936419 (89b:52020)

[B1]
H. Busemann, Volume in terms of concurrent cross-sections, Pacific J. Math. 3 (1953), 1-12. MR 0055712 (14:1115e)

[B2]
-, 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, New York, 1991. MR 1107516 (92c:52001)

[ES]
H. Edelsbrunner and S. S. Skiena, Probing convex bodies with X-rays, SIAM J. Comput. 17 (1988), 870-882. MR 961045 (89i:52002)

[F]
P. Funk, Über Flächen mit lauter geschlossenen geodätischen Linien, Math. Ann. 74 (1913), 278-300. MR 1511763

[G1]
R. J. Gardner, Intersection bodies and the Busemann-Petty problem, Trans. Amer. Math. Soc. 342 (1994), 435-445 (to appear). MR 1201126 (94e:52008)

[G2]
-, A positive answer to the Busemann-Petty problem in three dimensions, Ann. of Math. (2) (to appear). MR 1298719 (95i:52005)

[GV]
R. J. Gardner and A. Volčič, Tomography of convex and star bodies, Adv. Math, (to appear). MR 1296519 (95j:52013)

[Gi]
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)

[Gie]
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).

[GW]
P. R. Goodey and W. Weil, Zonoids and generalizations, Handbook of Convex Geometry (P. M. Gruber and J. M. Wills, eds.), North-Holland, Amsterdam, 1993, pp. 1297-1326.

[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)

[K]
V. L. Klee, Ungelöstes Problem Nr. 44, Elem. Math. 17 (1962), 84.

[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]
E. Lutwak, Intersection bodies and dual mixed volumes, Adv. Math. 71 (1988), 232-261. MR 963487 (90a:52023)

[M]
M. Meyer, On a problem of Busemann and Petty (to appear).

[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, 1989, pp. 64-104. MR 1008717 (90g:52003)

[P]
M. Papadimitrakis, On the Busemann-Petty problem about convex, centrally symmetric bodies in $             {\mathbb{R}^n}$, Mathematika 39 (1992), 258-266. MR 1203282 (94a:52019)

[S]
R. Schneider, Convex bodies: The Brunn-Minkowski Theory, Cambridge Univ. Press, Cambridge, 1993. MR 1216521 (94d:52007)

[SW]
R. Schneider and W. Weil, Zonoids and related topics, 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)

[W]
W. Weil, Stereology: A survey for geometers, Convexity and Its Applications (P. M. Gruber and J. M. Wills, eds.), Birkhäuser, Basel, 1983, pp. 360-412. MR 731118 (85e:52007)

[Z1]
Gaoyong Zhang, Intersection bodies and the four-dimensional Busemann-Petty problem, Duke Math. J. 71 (1993), 223-240. MR 1230300 (94f:52007)

[Z2]
-, Intersection bodies and the Busemann-Petty inequalities in $ {\mathbb{R}^4}$, Ann. of Math. (2) (to appear).

Similar Articles:

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 52A38, 52A40

Retrieve articles in all Journals with MSC (2000): 52A38, 52A40


Additional Information:

DOI: 10.1090/S0273-0979-1994-00493-8
PII: S 0273-0979(1994)00493-8
Keywords: Busemann-Petty problem, convex body, star body, section, intersection body, spherical Radon transform, Schwarz symmetral, geometric tomography
Copyright of article: Copyright 1994, American Mathematical Society




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia