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Blaschke- and Minkowski-endomorphisms of convex bodies


Author: Markus Kiderlen
Journal: Trans. Amer. Math. Soc. 358 (2006), 5539-5564
MSC (2000): Primary 52A20, 08A35; Secondary 43A90, 46T30
DOI: https://doi.org/10.1090/S0002-9947-06-03914-6
Published electronically: July 20, 2006
MathSciNet review: 2238926
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Abstract: We consider maps of the family of convex bodies in Euclidean $ d$-dimensional space into itself that are compatible with certain structures on this family: A Minkowski-endomorphism is a continuous, Minkowski-additive map that commutes with rotations. For $ d\ge 3$, a representation theorem for such maps is given, showing that they are mixtures of certain prototypes. These prototypes are obtained by applying the generalized spherical Radon transform to support functions. We give a complete characterization of weakly monotonic Minkowski-endomorphisms. A corresponding theory is developed for Blaschke-endomorphisms, where additivity is now understood with respect to Blaschke-addition. Using a special mixed volume, an adjoining operator can be introduced. This operator allows one to identify the class of Blaschke-endomorphisms with the class of weakly monotonic, non-degenerate and translation-covariant Minkowski-endomorphisms.

The following application is also shown: If a (weakly monotonic and) non-trivial endomorphism maps a convex body to a homothet of itself, then this body must be a ball.


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

  • 1. BERG, C. (1969) Corps convexes et potentiels sphériques, Danske Vid. Selsk. Mat.-Fys. Medd. 37, 6. MR 0254789 (40:7996)
  • 2. DUNKL, C.F. (1966) Operators and harmonic analysis on the sphere, Trans. Amer. Math. Soc. 125, 250-263. MR 0203371 (34:3224)
  • 3. ERDÉLYI, A., MAGNUS, W., OBERHETTINGER F., TRICOMI, F.G. (1953) Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York.
  • 4. FALLERT, H., GOODEY, P., WEIL, W. (1997) Spherical projections and centrally symmetric sets, Adv. Math. 129, No. 2, 301-322. MR 1462736 (98j:52005)
  • 5. GROEMER, H. (1996) Geometric Applications of Fourier Series and Spherical Harmonics, Encyclopedia of Mathematics and its Applications, Vol. 61, Cambridge University Press, Cambridge. MR 1412143 (97j:52001)
  • 6. GOODEY, P. (1998) Minkowski sums of projections of convex bodies, Mathematika 45, 253-268. MR 1695718 (2000e:52002)
  • 7. GOODEY, P., WEIL, W. (1993) Zonoids and generalizations, in: Handbook of convex geometry, eds. P.M. Gruber and J.M. Wills, Elsvier Sci. Publ. B.V., 1297-1326. MR 1243010 (95g:52015)
  • 8. GOODEY, P., JIANG, W. (2000) Minkowski sums of three dimensional projections of convex bodies, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 65, 105-119. MR 1809148 (2001j:52005)
  • 9. GOODEY, P., KIDERLEN, M., WEIL, W. (1998) Section and projection means of convex bodies, Mh. Math. 126, 37-54. MR 1633259 (99d:52004)
  • 10. HUG, D, LAST, G. (2000) On support measures in Minkowski spaces and contact distributions in stochastic geometry, Ann. of Probab. 28, No. 2, 796-850. MR 1782274 (2001g:60023)
  • 11. KIDERLEN, M. (1999) Schnittmittelungen und äquivariante Endomorphismen konvexer Körper, Ph.D. Thesis, University of Karlsruhe.
  • 12. KIDERLEN, M. (2004) Determination of a convex body from Minkowski sums of its projections, J. London Math. Soc. 70, No. 2, 529-544. MR 2078909
  • 13. SEELEY, R.T. (1966) Spherical harmonics, Amer. Math. Monthly 73, 115-121. MR 0201695 (34:1577)
  • 14. SCHNEIDER, R. (1969) Functions on a sphere with vanishing integrals over certain subspheres, J. Math. Anal. Appl. 26, 381-384. MR 0237723 (38:6004)
  • 15. SCHNEIDER, R. (1974) Equivariant endomorphisms of the space of convex bodies, Trans. Am. Math. Soc. 194, 53-78. MR 0353147 (50:5633)
  • 16. SCHNEIDER, R. (1974) Bewegungsäquivariante, additive und stetige Transformationen konvexer Bereiche, Arch. Math. 25, 303-312. MR 0344999 (49:9738)
  • 17. SCHNEIDER, R. (1977) Rekonstruktion eines konvexen Körpers aus seinen Projektionen, Math. Nachr. 79, 325-329. MR 0500538 (58:18147)
  • 18. SCHNEIDER, R. (1993) Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, Vol. 44, Cambridge University Press, Cambridge. MR 1216521 (94d:52007)
  • 19. SCHNEIDER, R., WEIL, W. (1992) Integralgeometrie, Teubner, Stuttgart. MR 1203777 (94c:52003)
  • 20. SCHWARTZ, L. (1957) Théorie des distributions, Hermann, Paris. MR 0209834 (35:730)
  • 21. SPRIESTERSBACH, K.K. (1998) Determination of a convex body from the average of projections and stability results, Math. Proc. Camb. Philos. Soc. 123, 561-569. MR 1608001 (99b:52010)
  • 22. WEIL, W. (1976) Centrally symmetric convex bodies and distributions, Israel J. Math. 24, 352-367. MR 0420436 (54:8450)
  • 23. WEIL, W. (1997) On the mean shape of particle processes, Adv. Appl. Probab. 29, 890-908. MR 1484773 (99b:60015)

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Additional Information

Markus Kiderlen
Affiliation: Mathematisches Institut II, Universität Karlsruhe, D-76128 Karlsruhe, Germany
Address at time of publication: Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Bgn. 1530, DK-8000 Aarhus C, Denmark
Email: kiderlen@imf.au.dk

DOI: https://doi.org/10.1090/S0002-9947-06-03914-6
Keywords: Convex body, Minkowski-addition, Blaschke-addition, Hausdorff metric, Steiner point, mixed volume, equivariant map, monotonic, spherical distribution, generalized function, spherical harmonic, multiplier
Received by editor(s): May 6, 2004
Received by editor(s) in revised form: November 22, 2004
Published electronically: July 20, 2006
Additional Notes: The majority of this note is taken from the author’s doctoral thesis [11], written in German at the University of Karlsruhe, Germany.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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