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

• Journal title : Honam Mathematical Journal
• Volume 34, Issue 3,  2012, pp.403-408
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2012.34.3.403
Title & Authors
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.; Lee, Young-Soo;

Abstract
In this paper, we prove that if K is a convex body in $\small{E^n}$ and $\small{E_i}$ and $\small{E_o}$ are inscribed ellipsoid and circumscribed ellipsoid of K respectively with $\small{{\alpha}E_i=E_o}$, then $\small{$({\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, $\small{{\omega}^2_n=V(K)V({\Gamma}_pK)}$ 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;
Language
English
Cited by
References
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