Intersection bodies and the Busemann-Petty problem

Gardner, R. J.

doi:10.1090/S0002-9947-1994-1201126-7

Intersection bodies and the Busemann-Petty problem
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Trans. Amer. Math. Soc. 342 (1994), 435-445 Request permission

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.

References

Keith Ball, Some remarks on the geometry of convex sets, Geometric aspects of functional analysis (1986/87), Lecture Notes in Math., vol. 1317, Springer, Berlin, 1988, pp. 224–231. MR 950983, DOI 10.1007/BFb0081743
Marcel Berger, Convexity, Amer. Math. Monthly 97 (1990), no. 8, 650–678. MR 1072810, DOI 10.2307/2324573
J. Bourgain, On the Busemann-Petty problem for perturbations of the ball, Geom. Funct. Anal. 1 (1991), no. 1, 1–13. MR 1091609, DOI 10.1007/BF01895416
J. Bourgain and J. Lindenstrauss, Projection bodies, Geometric aspects of functional analysis (1986/87), Lecture Notes in Math., vol. 1317, Springer, Berlin, 1988, pp. 250–270. MR 950986, DOI 10.1007/BFb0081746
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, DOI 10.1007/978-3-662-07441-1
Herbert Busemann, Volume in terms of concurrent cross-sections, Pacific J. Math. 3 (1953), 1–12. MR 55712
Herbert Busemann, Volumes and areas of cross-sections, Amer. Math. Monthly 67 (1960), 248–250. MR 120562, DOI 10.2307/2309685
H. Busemann and C. M. Petty, Problems on convex bodies, Math. Scand. 4 (1956), 88–94. MR 84791, DOI 10.7146/math.scand.a-10457
Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy, Unsolved problems in geometry, Problem Books in Mathematics, Springer-Verlag, New York, 1991. Unsolved Problems in Intuitive Mathematics, II. MR 1107516, DOI 10.1007/978-1-4612-0963-8
P. Funk, Über Flächen mit lauter geschlossenen geodätischen Linien, Math. Ann. 74 (1913), no. 2, 278–300 (German). MR 1511763, DOI 10.1007/BF01456044
R. J. Gardner, A positive answer to the Busemann-Petty problem in three dimensions, Ann. of Math. (2) 140 (1994), no. 2, 435–447. MR 1298719, DOI 10.2307/2118606
Apostolos A. Giannopoulos, A note on a problem of H. Busemann and C. M. Petty concerning sections of symmetric convex bodies, Mathematika 37 (1990), no. 2, 239–244. MR 1099772, DOI 10.1112/S002557930001295X
M. Giertz, A note on a problem of Busemann, Math. Scand. 25 (1969), 145–148 (1970). MR 262929, DOI 10.7146/math.scand.a-10951

Functional analytic characterizations of classes of convex bodies

Eric L. Grinberg, Spherical harmonics and integral geometry on projective spaces, Trans. Amer. Math. Soc. 279 (1983), no. 1, 187–203. MR 704609, DOI 10.1090/S0002-9947-1983-0704609-1
Eric L. Grinberg and Igor Rivin, Infinitesimal aspects of the Busemann-Petty problem, Bull. London Math. Soc. 22 (1990), no. 5, 478–484. MR 1082020, DOI 10.1112/blms/22.5.478
H. Hadwiger, Radialpotenzintegrale zentralsymmetrischer Rotations-körper und Ungleichheitsaussagen Busemannscher Art, Math. Scand. 23 (1969), 193–200 (1969) (German). MR 254739, DOI 10.7146/math.scand.a-10911
Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
Sigurdur Helgason, The totally-geodesic Radon transform on constant curvature spaces, Integral geometry and tomography (Arcata, CA, 1989) Contemp. Math., vol. 113, Amer. Math. Soc., Providence, RI, 1990, pp. 141–149. MR 1108651, DOI 10.1090/conm/113/1108651

Ungelöstes Problem Nr.

17

Foundations of differential geometry

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), no. 2, 164–175. MR 390914, DOI 10.1112/S0025579300006033
Erwin Lutwak, Intersection bodies and dual mixed volumes, Adv. in Math. 71 (1988), no. 2, 232–261. MR 963487, DOI 10.1016/0001-8708(88)90077-1
Erwin Lutwak, On some ellipsoid formulas of Busemann, Furstenberg and Tzkoni, Guggenheimer, and Petty, J. Math. Anal. Appl. 159 (1991), no. 1, 18–26. MR 1119418, DOI 10.1016/0022-247X(91)90218-O
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 (1987–88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 64–104. MR 1008717, DOI 10.1007/BFb0090049
V. I. Oliker, Hypersurfaces in $\textbf {R}^{n+1}$ with prescribed Gaussian curvature and related equations of Monge-Ampère type, Comm. Partial Differential Equations 9 (1984), no. 8, 807–838. MR 748368, DOI 10.1080/03605308408820348
Michael Papadimitrakis, On the Busemann-Petty problem about convex, centrally symmetric bodies in $\mathbf R^n$, Mathematika 39 (1992), no. 2, 258–266. MR 1203282, DOI 10.1112/S0025579300014996
C. M. Petty, Centroid surfaces, Pacific J. Math. 11 (1961), 1535–1547. MR 133733
Rolf Schneider, Functions on a sphere with vanishing integrals over certain subspheres, J. Math. Anal. Appl. 26 (1969), 381–384. MR 237723, DOI 10.1016/0022-247X(69)90160-7
Rolf Schneider, Smooth approximation of convex bodies, Rend. Circ. Mat. Palermo (2) 33 (1984), no. 3, 436–440. MR 779946, DOI 10.1007/BF02844505
Rolf Schneider and Wolfgang Weil, Zonoids and related topics, Convexity and its applications, Birkhäuser, Basel, 1983, pp. 296–317. MR 731116
Shukichi Tanno, Central sections of centrally symmetric convex bodies, Kodai Math. J. 10 (1987), no. 3, 343–361. MR 929994, DOI 10.2996/kmj/1138037465

Centered bodies and dual mixed volumes

Gao Yong Zhang, Intersection bodies and the four-dimensional Busemann-Petty problem, Internat. Math. Res. Notices 7 (1993), 233–240. MR 1230300, DOI 10.1155/S107379289300025X

Intersection bodies and the Busemann-Petty inequalities in