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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On a local version of the Aleksandrov-Fenchel inequality for the quermassintegrals of a convex body


Authors: A. Giannopoulos, M. Hartzoulaki and G. Paouris
Journal: Proc. Amer. Math. Soc. 130 (2002), 2403-2412
MSC (1991): Primary 52A20; Secondary 52A39, 52A40
Published electronically: January 23, 2002
MathSciNet review: 1897466
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Abstract: We discuss the analogue in the Brunn-Minkowski theory of the inequalities of Marcus-Lopes and Bergstrom about symmetric functions of positive reals and determinants of symmetric positive matrices respectively. We obtain a local version of the Aleksandrov-Fenchel inequality $W_i^2\geq W_{i-1}W_{i+1}$ which relates the quermassintegrals of a convex body $K$ to those of an arbitrary hyperplane projection of $K$. A consequence is the following fact: for any convex body $K$, for any $(n-1)$-dimensional subspace $E$ of ${\mathbb R}^n$ and any $t>0$,

\begin{displaymath}\frac{\vert P_E(K)+tD_E\vert}{\vert P_E(K)\vert}\leq\frac{\vert K+2tD_n\vert}{\vert K\vert},\end{displaymath}

where $D$ denotes the Euclidean unit ball and $\vert\cdot \vert$denotes volume in the appropriate dimension.


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Additional Information

A. Giannopoulos
Affiliation: Department of Mathematics, University of Crete, Iraklion, Greece
Email: apostolo@math.uch.gr

M. Hartzoulaki
Affiliation: Department of Mathematics, University of Crete, Iraklion, Greece
Email: hmarian@math.uch.gr

G. Paouris
Affiliation: Department of Mathematics, University of Crete, Iraklion, Greece
Email: paouris@math.uch.gr

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06329-3
PII: S 0002-9939(02)06329-3
Keywords: Mixed volumes, Aleksandrov-Fenchel inequality
Received by editor(s): December 20, 2000
Received by editor(s) in revised form: March 16, 2001
Published electronically: January 23, 2002
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2002 American Mathematical Society