Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

 
 

 

Zonoids with minimal volume-product--a new proof


Authors: Y. Gordon, M. Meyer and S. Reisner
Journal: Proc. Amer. Math. Soc. 104 (1988), 273-276
MSC: Primary 52A40; Secondary 52A20
DOI: https://doi.org/10.1090/S0002-9939-1988-0958082-9
MathSciNet review: 958082
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A new and simple proof of the following result is given: The product of the volumes of a symmetric zonoid $ A$ in $ {{\mathbf{R}}^n}$ and of its polar body is minimal if and only if $ A$ is the Minkowski sum of $ n$ segments.


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

  • [1] J. Bourgain and V. Milman, New volume ratio properties for convex symmetric bodies in $ {R^n}$, Invent. Math. 88 (1987), 319-340. MR 880954 (88f:52013)
  • [2] K. Leichtweiss, Konvexe Mengen, Springer-Verlag, Berlin-Heidelberg-New York, 1980. MR 586235 (81j:52001)
  • [3] M. Meyer, Une caracterisation volumique de certains espaces normés de dimension finie, Israel J. Math. 55 (1986), 317-326. MR 876398 (88f:52017)
  • [4] -, Sur le produit volumique de certain éspaces normés, Sem. d'Initiation à l'Analyse, 25 ème année, 1985-1986, no. 4.
  • [5] S. Reisner, Random polytopes and the volume product of symmetric convex bodies, Math. Scand. 57 (1985), 386-392. MR 832364 (87g:52011)
  • [6] -, Zonoids with minimal volume product, Math. Z. 192 (1986), 339-346. MR 845207 (87g:52022)
  • [7] -, Minimal volume product in Banach spaces with a $ 1$-unconditional basis, J. London Math. Soc. (2) 36 (1987), 126-136. MR 897680 (88h:46029)
  • [8] J. Saint-Raymond, Sur le volume des corps convexes symétriques, Sem. d'Initiation à l'Analyse, 20 ème année, 1980-1981, no. 11. MR 670798 (84j:46033)
  • [9] L. A. Santaló, Un invariante afin para los cuerpos convexos del espacio de $ n$ dimensiones, Portugal. Math. 8 (1949), 155-161.
  • [10] R. Schneider and W. Weil, Zonoids and selected topics, Convexity and its Applications, (P. M. Gruber and J. M. Wills, eds.), Birkhauser, Boston, Mass., 1983. MR 731116 (85c:52010)
  • [11] W. Marshall, I. Olkin and F. Proschan, Monotonicity of ratios of means and other applications of majorization, Inequalities (O. Shisha, ed.), Academic Press, New York, 1967. MR 0237727 (38:6008)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 52A40, 52A20

Retrieve articles in all journals with MSC: 52A40, 52A20


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0958082-9
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society