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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Stability results for first order projection bodies
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by P. R. Goodey and H. Groemer PDF
Proc. Amer. Math. Soc. 109 (1990), 1103-1114 Request permission

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.
References
    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)
  • Christian Berg, Corps convexes et potentiels sphériques, Mat.-Fys. Medd. Danske Vid. Selsk. 37 (1969), no. 6, 64 pp. (1969) (French). MR 254789
    • T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, Chelsea, New York, 1977.
  • 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
  • Stefano Campi, Reconstructing a convex surface from certain measurements of its projections, Boll. Un. Mat. Ital. B (6) 5 (1986), no. 3, 945–959 (English, with Italian summary). MR 871707
  • Paul R. Goodey, Instability of projection bodies, Geom. Dedicata 20 (1986), no. 3, 295–305. MR 845424, DOI 10.1007/BF00149579
  • H. Groemer, Stability properties of geometric inequalities, Amer. Math. Monthly 97 (1990), no. 5, 382–394. MR 1048910, DOI 10.2307/2324388
  • 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
    • T. Kubota, Über die konvex-geschlossenen Mannigfaltigkeiten im $n$-dimensionalen Raum, Tôhoku Univ. Sci. Rep. 14 (1925), 85-88.
  • Kurt Leichtweiss, Konvexe Mengen, Hochschultext [University Textbooks], Springer-Verlag, Berlin-New York, 1980 (German). MR 586235
    • H. Minkowski, Über die Körper konstanter Breite, Teubner, Leipzig-Berlin, 1911, pp. 277-279.
  • Claus Müller, Spherical harmonics, Lecture Notes in Mathematics, vol. 17, Springer-Verlag, Berlin-New York, 1966. MR 0199449
  • 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, 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, Functional equations connected with rotations and their geometric applications, Enseign. Math. (2) 16 (1970), 297–305 (1971). MR 287438
  • Rolf Schneider, Gleitkörper in konvexen Polytopen, J. Reine Angew. Math. 248 (1971), 193–220 (German). MR 279690, DOI 10.1515/crll.1971.248.193
  • R. Schneider, Stability in the Aleksandrov-Fenchel-Jessen theorem, Mathematika 36 (1989), no. 1, 50–59. MR 1014200, DOI 10.1112/S0025579300013565
  • Rolf Schneider and Wolfgang Weil, Zonoids and related topics, Convexity and its applications, Birkhäuser, Basel, 1983, pp. 296–317. MR 731116
  • Richard A. Vitale, $L_p$ metrics for compact, convex sets, J. Approx. Theory 45 (1985), no. 3, 280–287. MR 812757, DOI 10.1016/0021-9045(85)90051-6
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Additional Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 109 (1990), 1103-1114
  • MSC: Primary 52A40
  • DOI: https://doi.org/10.1090/S0002-9939-1990-1015678-5
  • MathSciNet review: 1015678