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Stability results for first order projection bodies

Authors: P. R. Goodey and H. Groemer
Journal: Proc. Amer. Math. Soc. 109 (1990), 1103-1114
MSC: Primary 52A40
MathSciNet review: 1015678
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Abstract: The motivation for this work comes from a result of Minkowski. He showed that if a three-dimensional convex body has the property that all its projections have the same perimeter, then the original body has constant width. Our objective was to extend this to a stability result and not to restrict ourselves to dimension three. The result we obtained shows that if two centrally symmetric bodies have projections which all have approximately the same mean width, then the two bodies are approximately the same up to translation. This is, in effect, a continuity result for the inverse of the spherical Radon transform. It is closely related to recent three-dimensional results of Campi and to work of Bourgain and Lindenstrauss, who consider the volumes of projections rather than their mean widths. The techniques we employ are drawn from the theory of spherical harmonics and from the theory of mixed volumes.

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  • [1] A. D. Aleksandrov, Zur Theorie der gemischten Volumina von konvexen Körpern II. Neue Ungleichungen Zwischen den gemischten Volumina und ihre Anwendungen, Mat. Sb. (N.S.) 2 (1937), 1205-1238. (Russian)
  • [2] C. Berg, Corps convexes et potentiels sphériques, Danske Vid. Selsk. Mat-fys. Medd. 37 (1969), 1-64. MR 0254789 (40:7996)
  • [3] T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, Chelsea, New York, 1977.
  • [4] J. Bourgain and J. Lindenstrauss, Projection bodies, Geometrie Aspects of Functional Analysis (J. Lindenstrauss and V. D. Milman, eds.) Lecture Notes in Math., vol. 1317, Springer-Verlag, New York, 1988, pp. 250-270. MR 950986 (89g:46024)
  • [5] S. Campi, Reconstructing a convex surface from certain measurements of its projection, Boll. Un. Mat. Ital. B 5 (1986), 945-959. MR 871707 (88f:52004)
  • [6] P. Goodey, Instability of projection bodies, Geom. Dedicata 20 (1986), 295-305. MR 845424 (87h:52007)
  • [7] H. Groemer, Stability properties of geometric inequalities, Amer. Math. Monthly (to appear). MR 1048910 (91a:52008)
  • [8] S. Helgason, Groups and geometric analysis, Academic Press, Orlando, FL, 1984. MR 754767 (86c:22017)
  • [9] T. Kubota, Über die konvex-geschlossenen Mannigfaltigkeiten im $ n$-dimensionalen Raum, Tôhoku Univ. Sci. Rep. 14 (1925), 85-88.
  • [10] K. Leichtweiss, Konvexe Mengen, Springer, Berlin, 1980. MR 586235 (81j:52001)
  • [11] H. Minkowski, Über die Körper konstanter Breite, Teubner, Leipzig-Berlin, 1911, pp. 277-279.
  • [12] C. Müller, Spherical harmonics, Lecture Notes in Math., vol. 17, Springer, Berlin, 1966. MR 0199449 (33:7593)
  • [13] R. Schneider, Zu einem Problem von Shephard über die Projektionen konvexer Körper, Math. Z. 101 (1967), 71-82. MR 0218976 (36:2059)
  • [14] -, Functions on a sphere with vanishing integrals over certain subspheres, J. Math. Anal. Appl. 26 (1969), 381-384. MR 0237723 (38:6004)
  • [15] -, Functional equations connected with rotations and their geometric applications, Enseign. Math. (2) 6 (1970), 297-305. MR 0287438 (44:4642)
  • [16] -, Gleitkörper in konvexen Polytopen, J. Reine Angew. Math. 248 (1971), 193-220. MR 0279690 (43:5411)
  • [17] -, Stability in the Aleksandrov-Fenchel-Jessen theorem, Mathematika 36 (1989), 50-59. MR 1014200 (90h:52015)
  • [18] R. Schneider and W. Weil, Zonoids and related topics, Convexity and its Applications (P. M. Gruber and J. M. Wills, eds.) Birkhäuser, Basel, 1983. MR 731116 (85c:52010)
  • [19] R. Vitale, $ {L_p}$ metrics for compact convex sets, J. Approx. Theory 45 (1985), 280-287. MR 812757 (88a:52003)

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