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Intersections of dilatates of convex bodies

Authors: Stefano Campi, Richard J. Gardner and Paolo Gronchi
Journal: Trans. Amer. Math. Soc. 364 (2012), 1193-1210
MSC (2010): Primary 52A20, 52A40; Secondary 52A38
Published electronically: October 25, 2011
MathSciNet review: 2869174
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Abstract: We initiate a systematic investigation into the nature of the function $ \alpha _K(L,\rho )$ that gives the volume of the intersection of one convex body $ K$ in $ \mathbb{R}^n$ and a dilatate $ \rho L$ of another convex body $ L$ in $ \mathbb{R}^n$, as well as the function $ \eta _K(L,\rho )$ that gives the $ (n-1)$-dimensional Hausdorff measure of the intersection of $ K$ and the boundary $ \partial (\rho L)$ of $ \rho L$. The focus is on the concavity properties of $ \alpha _K(L,\rho )$. Of particular interest is the case when $ K$ and $ L$ are symmetric with respect to the origin. In this situation, there is an interesting change in the concavity properties of $ \alpha _K(L,\rho )$ between dimension 2 and dimensions 3 or higher. When $ L$ is the unit ball, an important special case with connections to E. Lutwak's dual Brunn-Minkowski theory, we prove that this change occurs between dimension 2 and dimensions 4 or higher, and conjecture that it occurs between dimension 3 and dimension 4. We also establish an isoperimetric inequality with equality condition for subsets of equatorial zones in the sphere $ S^2$, and apply this and the Brunn-Minkowski inequality in the sphere to obtain results related to this conjecture, as well as to the properties of a new type of symmetral of a convex body, which we call the equatorial symmetral.

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Additional Information

Stefano Campi
Affiliation: Dipartimento di Ingegneria dell’Informazione, Università degli Studi di Siena, Via Roma 56, 53100 Siena, Italy

Richard J. Gardner
Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225-9063

Paolo Gronchi
Affiliation: Dipartimento di Matematica e Applicazioni per l’Architettura, Università degli Studi di Firenze, Piazza Ghiberti 27, 50122 Firenze, Italy

Keywords: Convex body, intersection, dilatate, Brunn-Minkowski inequality, isoperimetric inequality, symmetral, ball, sphere
Received by editor(s): February 22, 2010
Published electronically: October 25, 2011
Additional Notes: The second author was supported in part by the U.S. National Science Foundation Grant DMS-0603307.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.