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)

 
 

 

Convolutions and multiplier transformations of convex bodies


Author: Franz E. Schuster
Journal: Trans. Amer. Math. Soc. 359 (2007), 5567-5591
MSC (2000): Primary 52A20; Secondary 52A40, 43A90
DOI: https://doi.org/10.1090/S0002-9947-07-04270-5
Published electronically: May 11, 2007
MathSciNet review: 2327043
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Rotation intertwining maps from the set of convex bodies in $ \mathbb{R}^n$ into itself that are continuous linear operators with respect to Minkowski and Blaschke addition are investigated. The main focus is on Blaschke-Minkowski homomorphisms. We show that such maps are represented by a spherical convolution operator. An application of this representation is a complete classification of all even Blaschke-Minkowski homomorphisms which shows that these maps behave in many respects similar to the well known projection body operator. Among further applications is the following result: If an even Blaschke-Minkowski homomorphism maps a convex body to a polytope, then it is a constant multiple of the projection body operator.


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

  • 1. C. Berg, Corps convexes et potentiels sphériques, Danske Vid. Selsk. Mat.-Fys. Medd. 37, 6 (1969), 1-64. MR 0254789 (40:7996)
  • 2. J. Bourgain and J. Lindenstrauss, Projection bodies, Geometric aspects of functional analysis (1986/87), Springer, Berlin (1988), 250-270. MR 950986 (89g:46024)
  • 3. E.D. Bolker, A class of convex bodies, Trans. Amer. Math. Soc. 145 (1969), 323-345. MR 0256265 (41:921)
  • 4. E.M. Bronshtein, Extremal H-convex bodies (in Russian), Sibirskii Mat. Zh. 20 (1979), 412-415. English Translation: Siberian Math. J. 20, 295-297. MR 530507 (80f:52002)
  • 5. R.J. Gardner, Geometric Tomography, Cambridge University Press, 1995. MR 1356221 (96j:52006)
  • 6. P. Goodey, R. Schneider, On the intermediate area functions of convex bodies, Math. Z. 173 (1980), 185-194. MR 583385 (81k:52010)
  • 7. P. Goodey and W. Weil, The determination of convex bodies from the mean of random sections, Math. Proc. Camb. Phil. Soc. 112 (1992), 419-430. MR 1171176 (93i:52008)
  • 8. P. Goodey and W. Weil, Zonoids and generalizations, Handbook of Convex Geometry (P.M. Gruber and J.M. Wills, eds.), North-Holland, Amsterdam, 1993, 1297-1326. MR 1243010 (95g:52015)
  • 9. E. Grinberg and G. Zhang, Convolutions, transforms, and convex bodies, Proc. London Math. Soc. (3) 78 (1999), 77-115. MR 1658156 (99m:52009)
  • 10. H. Groemer, Geometric Applications of Fourier Series and Spherical Harmonics, Cambridge University Press, 1996. MR 1412143 (97j:52001)
  • 11. H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer, Berlin, 1957. MR 0102775 (21:1561)
  • 12. D. Hug and R. Schneider, Stability results involving surface area measures of convex bodies, Rendiconti Del Circolo Matematica Di Palermo 70 (2002), 21-51. MR 1962583 (2004b:52004)
  • 13. M. Kiderlen, Blaschke- and Minkowski-Endomorphisms of convex bodies, Trans. Amer. Math. Soc., to appear. MR 2238926
  • 14. D.A. Klain and G.C.Rota, Introduction to geometric probability, Cambridge University Press, Cambridge, 1997. MR 1608265 (2001f:52009)
  • 15. M. Ludwig, Projection bodies and valuations, Adv. Math. 172 (2002), 158-168. MR 1942402 (2003j:52012)
  • 16. M. Ludwig, Minkowski valuations, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4191-4213. MR 2159706 (2006f:52005)
  • 17. E. Lutwak, Mixed projection inequalities, Trans. Amer. Math. Soc. 287 (1985), no. 1, 91-105. MR 766208 (86c:52015)
  • 18. E. Lutwak, On quermassintegrals of mixed projection bodies, Geom. Dedicata 33 (1990), 51-58. MR 1042624 (91b:52008)
  • 19. E. Lutwak, Inequalities for mixed projection bodies, Trans. Amer. Math. Soc. 339 (1993), no. 2, 901-916. MR 1124171 (93m:52011)
  • 20. P. McMullen, Continuous translation invariant valuations on the space of compact convex sets, Arch. Math. 34, 377-384. MR 593954 (81m:52013)
  • 21. P. McMullen, Valuations and dissections, Handbook of Convex Geometry, Vol. B (P.M. Gruber and J.M. Wills, eds.), North Holland, Amsterdam, 1993, 933-990. MR 1243000 (95f:52018)
  • 22. P. McMullen and R. Schneider, Valuations on convex bodies, Convexity and its applications (P.M. Gruber and J.M. Wills, eds.), Birkhäuser, 1983, 170-247. MR 731112 (85e:52001)
  • 23. C.M. Petty, Projection bodies, Proceedings, Coll. Convexity, Copenhagen, 1965, Kobenhavns Univ. Mat. Inst. (1967), 234-241. MR 0216369 (35:7203)
  • 24. C.M. Petty, Isoperimetric problems, Proc. Conf. on Convexity and Combinatorial Geometry, Univ. of Oklahoma, June 1971 (1972), 26-41. MR 0362057 (50:14499)
  • 25. R. Schneider, Zu einem Problem von Shephard über die Projektionen konvexer Körper, Math. Z. 101 (1967), 71-82. MR 0218976 (36:2059)
  • 26. R. Schneider, Equivariant endomorphisms of the space of convex bodies, Trans. Amer. Math. Soc. 194 (1974), 53-78. MR 0353147 (50:5633)
  • 27. R. Schneider, Bewegungsäquivariante, additive und stetige Transformationen konvexer Bereiche, Arch. Math. 25 (1974), 303-312. MR 0344999 (49:9738)
  • 28. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, 1993. MR 1216521 (94d:52007)
  • 29. F.E. Schuster, Volume Inequalities and Additive Maps of Convex Bodies, in preparation.
  • 30. W. Weil, Decomposition of convex bodies, Mathematika 21 (1974), 19-25. MR 0365359 (51:1611)
  • 31. W. Weil, Über den Vektorraum der Differenzen von Stützfunktionen konvexer Körper, Math. Nachr. 59 (1974), 353-369. MR 0341283 (49:6033)
  • 32. W. Weil, Kontinuierliche Linearkombination von Strecken, Math. Z. 148 (1976), 71-84. MR 0400052 (53:3887)
  • 33. W. Weil, On surface area measures of convex bodies, Geom. Dedicata 9 (1980), 299-306. MR 585937 (81m:52014)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 52A20, 52A40, 43A90

Retrieve articles in all journals with MSC (2000): 52A20, 52A40, 43A90


Additional Information

Franz E. Schuster
Affiliation: Institut für Diskrete Mathematik and Geometrie, Technische Universität Wien, Wiedner Hauptstrasse 8-10/1046, 1040 Wien, Austria
Email: fschuster@osiris.tuwien.ac.at

DOI: https://doi.org/10.1090/S0002-9947-07-04270-5
Keywords: Convex bodies, Minkowski addition, Blaschke addition, rotation intertwining map, spherical convolution, spherical harmonic, multiplier transformation, projection body, Petty conjecture
Received by editor(s): July 4, 2005
Received by editor(s) in revised form: December 7, 2005
Published electronically: May 11, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society