Centered bodies and dual mixed volumes

Zhang, Gao Yong

doi:10.1090/S0002-9947-1994-1254193-9

Centered bodies and dual mixed volumes
HTML articles powered by AMS MathViewer

by Gao Yong Zhang

Trans. Amer. Math. Soc. 345 (1994), 777-801

DOI: https://doi.org/10.1090/S0002-9947-1994-1254193-9

PDF | Request permission

Abstract:

We establish a number of characterizations and inequalities for intersection bodies, polar projection bodies and curvature images of projection bodies in ${{\mathbf {R}}^n}$ by using dual mixed volumes. One of the inequalities is between the dual Quermassintegrals of centered bodies and the dual Quermassintegrals of their central $(n - 1)$-slices. It implies Lutwak’s affirmative answer to the Busemann-Petty problem when the body with the smaller sections is an intersection body. We introduce and study the intersection body of order i of a star body, which is dual to the projection body of order i of a convex body. We show that every sufficiently smooth centered body is a generalized intersection body. We discuss a type of selfadjoint elliptic differential operator associated with a convex body. These operators give the openness property of the class of curvature functions of convex bodies. They also give an existence theorem related to a well-known uniqueness theorem about deformations of convex hypersurfaces in global differential geometry.

References

Thierry Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Springer-Verlag, New York, 1982. MR 681859, DOI 10.1007/978-1-4612-5734-9
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
Ethan D. Bolker, A class of convex bodies, Trans. Amer. Math. Soc. 145 (1969), 323–345. MR 256265, DOI 10.1090/S0002-9947-1969-0256265-X
T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, Springer-Verlag, Berlin-New York, 1974 (German). Berichtigter Reprint. MR 0344997
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
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, A theorem on convex bodies of the Brunn-Minkowski type, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 27–31. MR 28046, DOI 10.1073/pnas.35.1.27
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
Herbert Busemann, Convex surfaces, Interscience Tracts in Pure and Applied Mathematics, no. 6, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. MR 0105155
G. D. Chakerian, Sets of constant relative width and constant relative brightness, Trans. Amer. Math. Soc. 129 (1967), 26–37. MR 212678, DOI 10.1090/S0002-9947-1967-0212678-1
Shiu Yuen Cheng and Shing Tung Yau, On the regularity of the solution of the $n$-dimensional Minkowski problem, Comm. Pure Appl. Math. 29 (1976), no. 5, 495–516. MR 423267, DOI 10.1002/cpa.3160290504
R. J. Gardner, Intersection bodies and the Busemann-Petty problem, Trans. Amer. Math. Soc. 342 (1994), no. 1, 435–445. MR 1201126, DOI 10.1090/S0002-9947-1994-1201126-7
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
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443
P. R. Goodey, Centrally symmetric convex sets and mixed volumes, Mathematika 24 (1977), no. 2, 193–198. MR 487788, DOI 10.1112/S0025579300009104
Paul Goodey and Wolfgang Weil, Centrally symmetric convex bodies and the spherical Radon transform, J. Differential Geom. 35 (1992), no. 3, 675–688. MR 1163454
P. Goodey, E. Lutwak, and W. Weil, Functional analytic characterizations of classes of convex bodies, Math. Z. 222 (1996), no. 3, 363–381. MR 1400197, DOI 10.1007/PL00004539
Paul Goodey and Gao Yong Zhang, Characterizations and inequalities for zonoids, J. London Math. Soc. (2) 53 (1996), no. 1, 184–196. MR 1362695, DOI 10.1112/jlms/53.1.184
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
Philip Hartman and Aurel Wintner, On the third fundamental form of a surface, Amer. J. Math. 75 (1953), 298–334. MR 55734, DOI 10.2307/2372455
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
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, Volume of mixed bodies, Trans. Amer. Math. Soc. 294 (1986), no. 2, 487–500. MR 825717, DOI 10.1090/S0002-9947-1986-0825717-3
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, Centroid bodies and dual mixed volumes, Proc. London Math. Soc. (3) 60 (1990), no. 2, 365–391. MR 1031458, DOI 10.1112/plms/s3-60.2.365
Erwin Lutwak, Mixed affine surface area, J. Math. Anal. Appl. 125 (1987), no. 2, 351–360. MR 896173, DOI 10.1016/0022-247X(87)90097-7
Erwin Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975), no. 2, 531–538. MR 380631

Selected affine isoperimetric inequalities

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
Louis Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1953), 337–394. MR 58265, DOI 10.1002/cpa.3160060303
V. I. Oliker, On certain elliptic differential equations on a hypersphere and their geometric applications, Indiana Univ. Math. J. 28 (1979), no. 1, 35–51. MR 523622, DOI 10.1512/iumj.1979.28.28003
V. I. Oliker, Infinitesimal deformations preserving parallel normal vector fields, Geometry and differential geometry (Proc. Conf., Univ. Haifa, Haifa, 1979), Lecture Notes in Math., vol. 792, Springer, Berlin, 1980, pp. 383–405. MR 585882

Some remarks on elliptic equations and infinitesimal deformations of submanifolds

C. M. Petty, Centroid surfaces, Pacific J. Math. 11 (1961), 1535–1547. MR 133733
C. M. Petty, Projection bodies, Proc. Colloquium on Convexity (Copenhagen, 1965) Kobenhavns Univ. Mat. Inst., Copenhagen, 1967, pp. 234–241. MR 0216369
Aleksey Vasil′yevich Pogorelov, The Minkowski multidimensional problem, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto-London, 1978. Translated from the Russian by Vladimir Oliker; Introduction by Louis Nirenberg. MR 0478079
Rolf Schneider, Zur einem Problem von Shephard über die Projektionen konvexer Körper, Math. Z. 101 (1967), 71–82 (German). MR 218976, DOI 10.1007/BF01135693
Rolf Schneider and Wolfgang Weil, Zonoids and related topics, Convexity and its applications, Birkhäuser, Basel, 1983, pp. 296–317. MR 731116
Rolf Schneider, On the Aleksandrov-Fenchel inequality, Discrete geometry and convexity (New York, 1982) Ann. New York Acad. Sci., vol. 440, New York Acad. Sci., New York, 1985, pp. 132–141. MR 809200, DOI 10.1111/j.1749-6632.1985.tb14547.x
J. T. Schwartz, Nonlinear functional analysis, Notes on Mathematics and its Applications, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Notes by H. Fattorini, R. Nirenberg and H. Porta, with an additional chapter by Hermann Karcher. MR 0433481
Laurent Schwartz, Théorie des distributions, Publications de l’Institut de Mathématique de l’Université de Strasbourg, IX-X, Hermann, Paris, 1966 (French). Nouvelle édition, entiérement corrigée, refondue et augmentée. MR 0209834
Robert S. Strichartz, $L^p$ estimates for Radon transforms in Euclidean and non-Euclidean spaces, Duke Math. J. 48 (1981), no. 4, 699–727. MR 782573, DOI 10.1215/S0012-7094-81-04839-0
Wolfgang Weil, Decomposition of convex bodies, Mathematika 21 (1974), 19–25. MR 365359, DOI 10.1112/S0025579300005738
Wolfgang Weil, Centrally symmetric convex bodies and distributions, Israel J. Math. 24 (1976), no. 3-4, 352–367. MR 420436, DOI 10.1007/BF02834765
Wolfgang Weil, Centrally symmetric convex bodies and distributions. II, Israel J. Math. 32 (1979), no. 2-3, 173–182. MR 531260, DOI 10.1007/BF02764913
Wolfgang Weil, On surface area measures of convex bodies, Geom. Dedicata 9 (1980), no. 3, 299–306. MR 585937, DOI 10.1007/BF00181175

Intersection bodies and Busemann-Petty inequalities in

No polytope is an intersection body

Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, vol. 94, Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1971 edition. MR 722297
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
Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521, DOI 10.1017/CBO9780511526282