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)



The Wills conjecture

Author: Noah Samuel Brannen
Journal: Trans. Amer. Math. Soc. 349 (1997), 3977-3987
MSC (1991): Primary 52A40
MathSciNet review: 1373630
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Two strengthenings of the Wills conjecture, an extension of Bonnesen's inradius inequality to $n$-dimensional space, are obtained. One is the sharpest of the known strengthenings of the conjecture in three dimensions; the other employs techniques which are fundamentally different from those utilized in the other proofs.

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

  • 1. A. D. Aleksandrov, On the theory of mixed volumes of convex bodies, Mat. Sb. 3 (1938), no. 45, 28-46. (Russian).
  • 2. J. Bokowski, Eine verschärfte ungleichung zwischen Volumen, Oberfläche und Inkugelradius im ${R}^m$, Elem. Math. 28 (1973), 43-44.MR 48:7119
  • 3. J. Bokowski and E. Heil, Integral representations of quermassintegrals and bonnesen-style inequalities, Arch. Math. 47 (1986), 79-89.MR 88b:52008
  • 4. G. Bol, Beweis einer Vermutung von H. Minkowski, Abh. Math. Sem. Univ. Hamburg 15 (1943), 37-56.MR 7:474f
  • 5. T. Bonnesen, Les problèmes des isopérimetres et des isépiphanes, Gauthier-Villars, Paris, 1929.
  • 6. G. D. Chakerian, Higher dimensional analogues of an isoperimetric inequality of Benson, Mathematische Nachrichten 48 (1971), 33-41. MR 44:4643
  • 7. V. I. Diskant, A generalization of Bonnesen's inequalities, Soviet Math. Doklady 14 (1973), no. 6, 1728-1731 (translation of Doklady Akad. Nauk SSSR 213 (1973), 519-521). MR 49:3688
  • 8. H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer, Berlin, 1957.MR 21:1561
  • 9. G. Matheron, La formule de Steiner pour les érosions, Journal of Applied Probability 15 (1978), 126-135. MR 58:12874
  • 10. R. Osserman, Bonnesen-style isoperimetric inequalities, American Math. Monthly 86 (1979), 1-29.MR 80h:52013
  • 11. J. R. Sangwine-Yager, Bonnesen-style inequalities for Minkowski relative geometry, Transactions of the American Mathematical Society 307 (1988), no. 1, 373-382.MR 89g:52007
  • 12. -, A Bonnesen-style inradius inequality in 3-space, Pacific Journal of Mathematics 134 (1988), 173-178.MR 89f:52032
  • 13. R. Schneider, Convex bodies: The Brunn-Minkowski theory, Cambridge University Press, Cambridge, 1993.MR 94d:52007
  • 14. B. Teissier, Bonnesen-type inequalities in algebraic geometry, Seminar on Differential Geometry (Princeton), Princeton University Press, 1982, pp. 85-105.MR 83d:52010
  • 15. J. M. Wills, Zum Verhaltnis von Volumen zur Oberfläche bei konvexen Körpern, Arch. Math 21 (1970), 557-560.MR 43:3923
  • 16. -, Minkowski's successive minima and the zeros of a convexity function, Monatsh. Math. 109 (1990), 157-164.MR 91f:52012

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 52A40

Retrieve articles in all journals with MSC (1991): 52A40

Additional Information

Noah Samuel Brannen
Affiliation: 2-36-6 Tama-cho, Fuchu-shi, Tokyo 183, Japan

Keywords: Circumradius, convex body, inner parallel body, inradius, mixed volume, quermassintegral.
Received by editor(s): February 9, 1995
Received by editor(s) in revised form: August 15, 1995
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society