Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Centered bodies and dual mixed volumes


Author: Gao Yong Zhang
Journal: Trans. Amer. Math. Soc. 345 (1994), 777-801
MSC: Primary 52A39
DOI: https://doi.org/10.1090/S0002-9947-1994-1254193-9
MathSciNet review: 1254193
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] T. Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations, Springer-Verlag, 1982. MR 681859 (85j:58002)
  • [2] K. Ball, Some remarks on the geometry of convex sets, Geometric Aspects of Functional Analysis, 1986-1987, Lecture Notes in Math., vol. 1317, Springer, 1988. MR 950983 (89h:52009)
  • [3] M. Berger, Convexity, Amer. Math. Monthly 97 (1990), 650-678. MR 1072810 (91f:52001)
  • [4] E. D. Bolker, A class of convex bodies, Trans. Amer. Math. Soc. 145 (1969), 323-345. MR 0256265 (41:921)
  • [5] T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, Springer-Verlag, Berlin, 1934. MR 0344997 (49:9736)
  • [6] J. Bourgain, On the Busemann-Petty problem for perturbations of ball, Geom. Funct. Anal. 1 (1991), 1-13. MR 1091609 (92c:52008)
  • [7] Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Springer-Verlag, Berlin, 1988. MR 936419 (89b:52020)
  • [8] H. Busemann, A theorem on convex bodies of the Brunn-Minkowski type, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 27-31. MR 0028046 (10:395c)
  • [9] H. Busemann and C. M. Petty, Problems on convex bodies, Math. Scand. 4 (1956), 88-94. MR 0084791 (18:922b)
  • [10] H. Busemann, Convex surfaces, Interscience, New York, 1958. MR 0105155 (21:3900)
  • [11] G. D. Chakerian, Sets of constant relative width and constant relative brightness, Trans. Amer. Math. Soc. 129 (1967), 26-37. MR 0212678 (35:3545)
  • [12] S. Y. Cheng and S. T. Yau, On the regularity of the solution of the n-dimensional Minkowski problem, Comm. Pure Appl. Math. 29 (1976), 495-516. MR 0423267 (54:11247)
  • [13] R. Gardner, Intersection bodies and the Busemann-Petty problem, Trans. Amer. Math. Soc. 342 (1994), 435-445. MR 1201126 (94e:52008)
  • [14] 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)
  • [15] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag (1977). MR 0473443 (57:13109)
  • [16] P. R. Goodey, Centrally symmetric convex sets and mixed volumes, Mathematika 24 (1977), 193-198. MR 0487788 (58:7393)
  • [17] P. Goodey and W. Weil, Centrally symmetric convex bodies and the spherical Radon transform, J. Differential Geometry 35 (1992), 675-688. MR 1163454 (93g:44005)
  • [18] P. Goodey, E. Lutwak and W. Weil, Functional analytic characterizations of classes of convex bodies, (to appear). MR 1400197 (97i:52002)
  • [19] P. Goodey and G. Zhang, Characterizations and inequalities of zonoids, (to appear). MR 1362695 (97c:52025)
  • [20] E. L. Grinberg and I. Rivin, Infinitesimal aspects of the Busemann and Petty problem, Bull. London Math. Soc. 22 (1990), 478-484. MR 1082020 (92e:52012)
  • [21] P. Hartman and A. Wintner, On the third fundamental form of a surface, Amer. J. Math. 75 (1953), 298-334. MR 0055734 (14:1119c)
  • [22] S. Helgason, Groups and geometric analysis, Academic Press, 1984. MR 754767 (86c:22017)
  • [23] D. G. Larman and C. A. Rogers, The existence of a centrally symmetric convex body with central cross-sections that are unexpectedly small, Mathematika 22 (1975), 164-175. MR 0390914 (52:11737)
  • [24] E. Lutwak, Volume of mixed bodies, Trans. Amer. Math. Soc. 294 (1986), 487-500. MR 825717 (87f:52017)
  • [25] -, Intersection bodies and dual mixed volumes, Adv. Math. 71 (1988), 232-261. MR 963487 (90a:52023)
  • [26] -, Centroid bodies and dual mixed volumes, Proc. London Math. Soc. (3) 60 (1990), 365-391. MR 1031458 (90k:52024)
  • [27] -, Mixed affine surface area, J. Math. Anal. Appl. 125 (1987), 351-360. MR 896173 (89d:52009)
  • [28] -, Dual mixed volumes, Pacific J. Math. 58 (1975), 531-538. MR 0380631 (52:1528)
  • [29] -, Selected affine isoperimetric inequalities, Convex Geometry (P. M. Gruberand J. M. Wills, eds.), Elsevier Science Publishers, 1993.
  • [30] 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)
  • [31] L. Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Comm. Pure Appl. Math. 6 (1953), 337-394. MR 0058265 (15:347b)
  • [32] V. I. Oliker, On certain elliptic differential equations on a hypersphere and their geometric applications, Indiana Univ. Math. J. 28 (1979), 35-51. MR 523622 (80i:53031)
  • [33] -, Infinitesimal deformations preserving parallel normal vector fields, Lecture Notes in Math., vol. 792, Springer, 1980, pp. 383-405. MR 585882 (82f:53071)
  • [34] -, Some remarks on elliptic equations and infinitesimal deformations of submanifolds, Lecture Notes in Math., vol. 838, Springer, 1981, pp. 211-220.
  • [35] C. M. Petty, Centroid bodies, Pacific J. Math. 11 (1961), 1535-1547. MR 0133733 (24:A3558)
  • [36] -, Projection bodies, Proc. Colloq. on Convexity (Copenhagen, 1965), Københavns. Univ. Mat. Inst. 1967, pp. 234-241. MR 0216369 (35:7203)
  • [37] A. V. Pogorelov, The Minkowski multidimensional problem, Wiley, New York, 1978. MR 0478079 (57:17572)
  • [38] R. Schneider, Zu einem Problem, Projektionen, Körper von Shephard über die projectionen konvexer körper, Math. Z. 101 (1967), 71-82. MR 0218976 (36:2059)
  • [39] 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)
  • [40] R. Schneider, On the Alexandrov-Fenchel inequality, Discrete Geometry and Convexity, (J. E. Goodman et al., eds.), vol. 440, Ann. New York Acad. Sci., 1985, pp. 132-141. MR 809200 (87c:52019)
  • [41] J. Schwartz, Nonlinear functional analysis, Gordon and Breach, New York, 1968. MR 0433481 (55:6457)
  • [42] L. Schwartz, Théorie des distributions, Hermann, Paris, 1966. MR 0209834 (35:730)
  • [43] R. Strichartz, $ {L^p}$ estimates for Radon transforms in Euclidean and non-Euclidean spaces, Duke Math. J. 48 (1981), 699-727. MR 782573 (86k:43008)
  • [44] W. Weil, Decomposition of convex bodies, Mathematika 21 (1974), 19-25. MR 0365359 (51:1611)
  • [45] -, Centrally symmetric convex bodies and distributions, Israel J. Math. 24 (1976), 352-367. MR 0420436 (54:8450)
  • [46] -, Centrally symmetric convex bodies and distributions II, Israel J. Math. 32 (1979), 173-182. MR 531260 (80g:52003)
  • [47] -, On surface area measure of convex bodies, Geom. Dedicata 9 (1980), 299-306. MR 585937 (81m:52014)
  • [48] G. Zhang, Intersection bodies and Busemann-Petty inequalities in $ {{\mathbf{R}}^4}$, Ann. of Math, (to appear).
  • [49] -, No polytope is an intersection body, (to appear).
  • [50] F. Warner, Foundations of differentiable manifolds and Lie groups, Springer-Verlag, 1983. MR 722297 (84k:58001)
  • [51] R. Gardner, A positive answer to the Busemann-Petty problem in three dimensions, Ann. of Math. (to appear). MR 1298719 (95i:52005)
  • [52] R. Schneider, Convex bodies: The Brunn-Minkowski theory, Cambridge Univ. Press, 1993. MR 1216521 (94d:52007)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 52A39

Retrieve articles in all journals with MSC: 52A39


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1254193-9
Keywords: Convex body, star body, intersection body, dual mixed volume, geometric inequality, curvature function
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society