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)

 
 

 

Equivariant endomorphisms of the space of convex bodies


Author: Rolf Schneider
Journal: Trans. Amer. Math. Soc. 194 (1974), 53-78
MSC: Primary 52A20; Secondary 53C65
DOI: https://doi.org/10.1090/S0002-9947-1974-0353147-1
MathSciNet review: 0353147
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider maps of the set of convex bodies in d-dimensional Euclidean space into itself which are linear with respect to Minkowski addition, continuous with respect to Hausdorff metric, and which commute with rigid motions. Examples constructed by means of different methods show that there are various nontrivial maps of this type. The main object of the paper is to find some reasonable additional assumptions which suffice to single out certain special maps, namely suitable combinations of dilatations and reflections, and of rotations if $ d = 2$. For instance, we determine all maps which, besides having the properties mentioned above, commute with affine maps, or are surjective, or preserve the volume. The method of proof consists in an application of spherical harmonics, together with some convexity arguments.


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

  • [1] Ch. Berg, Corps convexes et potentiels sphériques, Mat.-Fys. Medd. Danske Vid. Selsk. 37 (1969), no. 6, 64 pp. MR 40 #7996. MR 0254789 (40:7996)
  • [2] -, Abstract Steiner points for convex polytopes, J. London Math. Soc. (2) 4 (1971), 176-180. MR 45 #7593. MR 0298541 (45:7593)
  • [3] W. Blaschke, Kreis und Kugel, 2nd ed., de Gruyter, Berlin, 1956. MR 17, 1123. MR 0077958 (17:1123d)
  • [4] H. Boerner, Darstellungen von Gruppen. Mit Berücksichtigung der Bedürfnisse der modernen Physik, Zweite, überarbeitete Auflage, Die Grundlehren der math. Wissenschaften, Band 74, Springer-Verlag, Berlin and New York, 1967. MR 37 #5307. MR 0229733 (37:5307)
  • [5] T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, Springer, Berlin, 1934. MR 0344997 (49:9736)
  • [6] H. Busemann, Convex surfaces, Interscience Tracts in Pure and Appl. Math., no. 6, Interscience, New York, 1958. MR 21 #3900. MR 0105155 (21:3900)
  • [7] R. R. Coifman and G. Weiss, Representations of compact groups and spherical harmonics, Enseignement Math. (2) 14 (1968), 121-173. MR 41 #537. MR 0255877 (41:537)
  • [8] R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math., vol. 242, Springer, Berlin and New York, 1971. MR 0499948 (58:17690)
  • [9] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience, New York, 1965. MR 0065391 (16:426a)
  • [10] C. F. Dunkl, Operators and harmonic analysis on the sphere, Trans. Amer. Math. Soc. 125 (1966), 250-263. MR 34 #3224. MR 0203371 (34:3224)
  • [11] R. E. Edwards, On convex spans of translates of functions on a group, Proc. London Math. Soc. (3) 3 (1953), 222-242. MR 15, 101. MR 0056615 (15:101e)
  • [12] W. Fenchel and B. Jessen, Mengenfunktionen und konvexe Körper, Danske Vid. Selsk. Mat.-Fys. Medd. 16 (1938), no. 3, 31 pp. MR 0031113 (11:101h)
  • [13] B. Grünbaum, Measures of symmetry for convex bodies, Proc. Sympos. Pure Math., vol. 7, Amer. Math. Soc., Providence, R.I. 1963, pp. 233-270. MR 27 #6187.
  • [14] -, Convex polytopes, Pure and Appl. Math., vol. 16, Interscience, New York, 1967. MR 37 #2085.
  • [15] H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer, Berlin, 1957. MR 21 #1561.
  • [16] H. Hadwiger and R. Schneider, Vektorielle Integralgeometrie, Elem. Math. 26 (1971), 49-57. MR 44 #967. MR 0283737 (44:967)
  • [17] T. Kubota, Über die konvex-geschlossenen Mannigfaltigkeiten im n-dimensionalen Raume, Sci. Rep. Tôhoku Imperial Univ. 14 (1925), 85-99.
  • [19] C. Müller, Spherical harmonics, Lecture Notes in Math., vol. 17, Springer-Verlag, Berlin and New York, 1966. MR 33 #7593. MR 0199449 (33:7593)
  • [20] G. T. Sallee, A non-continuous ``Steiner point", Israel J. Math. 10 (1971), 1-5. MR 45 #5873. MR 0296814 (45:5873)
  • [21] H. Schaal, Prüfung einer Kreisform mit Hohlwinkel und Taster, Elem. Math. 17 (1962), 33-38. MR 25 #1489. MR 0138041 (25:1489)
  • [22] R. Schneider, On Steiner points of convex bodies, Israel J. Math. 9 (1971), 241-249. MR 43 #3918. MR 0278187 (43:3918)
  • [23] -, Gleitkörper in konvexen Polytopen, J. Reine Angew. Math. 248 (1971), 193-220. MR 43 #5411. MR 0279690 (43:5411)
  • [24] -, Krümmungsschwerpunkte konvexer Körper. II, Abh. Math. Sem. Univ. Hamburg 37 (1972), 204-217. MR 0331220 (48:9554)
  • [25] G. C. Shephard, Euler-type relations for convex polytopes, Proc. London Math. Soc. (3) 18 (1968), 597-606. MR 38 #606. MR 0232280 (38:606)
  • [26] -, A uniqueness theorem for the Steiner point of a convex region, J. London Math. Soc. 43 (1968), 439-444. MR 37 #3447. MR 0227863 (37:3447)
  • [27] G. Valette, Subadditive affine-invariant transformations of convex bodies, Geometriae Dedicata (to appear). MR 0375083 (51:11279)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 52A20, 53C65

Retrieve articles in all journals with MSC: 52A20, 53C65


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0353147-1
Keywords: Convex body, Minkowski addition, Hausdorff metric, motion group, equivariant map, support function, mean width, Steiner point, characterization of functionals defined on the set of convex bodies, quermassintegral, convex function, spherical harmonic, multiplier
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society