Browse > Article
http://dx.doi.org/10.5831/HMJ.2012.34.3.403

A NEW LOWER BOUND FOR THE VOLUME PRODUCT OF A CONVEX BODY WITH CONSTANT WIDTH AND POLAR DUAL OF ITS p-CENTROID BODY  

Chai, Y.D. (Department of Mathematics, Sungkyunkwan University)
Lee, Young-Soo (Department of Mathematics, Sungkyunkwan University)
Publication Information
Honam Mathematical Journal / v.34, no.3, 2012 , pp. 403-408 More about this Journal
Abstract
In this paper, we prove that if K is a convex body in $E^n$ and $E_i$ and $E_o$ are inscribed ellipsoid and circumscribed ellipsoid of K respectively with ${\alpha}E_i=E_o$, then $\[({\alpha})^{\frac{n}{p}+1}\]^n{\omega}^2_n{\geq}V(K)V({\Gamma}^{\ast}_pK){\geq}\[(\frac{1}{\alpha})^{\frac{n}{p}+1}\]^n{\omega}^2_n$. Lutwak and Zhang[6] proved that if K is a convex body, ${\omega}^2_n=V(K)V({\Gamma}_pK)$ if and only if K is an ellipsoid. Our inequality provides very elementary proof for their result and this in turn gives a lower bound of the volume product for the sets of constant width.
Keywords
Convex body; constant width; polar body; volume product; p-centroid body;
Citations & Related Records
연도 인용수 순위
  • Reference
1 G. D. Chakerian and H. Groemer, Convex Bodies of Constant Width, In: Convexity and Its Applications, ed. by P. M. Gruber and J. M. Wills. Birkhauser, Basel, 1983.
2 H.G. Eggleston, Convexity, Cambridge Univ. Press, 1958.
3 R. J. Gardner, Geometric Tomography, Cambridge Univ. Press, Cambridge, 1995.
4 H. Groemer, Stability Theorem for convex domains of constant width, Canad. Math. Bull. 31(1988), 328-337.   DOI
5 R. Howard, Convex bodies of constant width and constant brightness, Adv. Math. 204 (2006), no. 1, 241-261.   DOI   ScienceOn
6 E. Lutwak and G. Zhang, Blaschke-Santalo inequality, J. Differential Geom., Vol.47(1997), 1-16.   DOI
7 Z. A.Melzak, A note on sets of constant width. Proc. Amer. Math. Soc. 11 (1960) 493-497. Cambridge Univ. Press, Cambridge, 1993.
8 I.M.Yaglom and V.G. Boltyanskii, Convex Figures, Hol, Rinehart and Winston, New York, 1961.