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Bodies with similar projections


Authors: G. D. Chakerian and E. Lutwak
Journal: Trans. Amer. Math. Soc. 349 (1997), 1811-1820
MSC (1991): Primary 52A40
DOI: https://doi.org/10.1090/S0002-9947-97-01760-1
MathSciNet review: 1390034
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Abstract: Aleksandrov's projection theorem characterizes centrally symmetric convex bodies by the measures of their orthogonal projections on lower dimensional subspaces. A general result proved here concerning the mixed volumes of projections of a collection of convex bodies has the following corollary. If $K$ is a convex body in ${\mathbb {R}}^{n}$ whose projections on $r$-dimensional subspaces have the same $r$-dimensional volume as the projections of a centrally symmetric convex body $M$, then the Quermassintegrals satisfy $W_{j}(M)\ge W_{j}(K)$, for $0\le j < n-r$, with equality, for any $j$, if and only if $K$ is a translate of $M$. The case where $K$ is centrally symmetric gives Aleksandrov's projection theorem.


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Additional Information

G. D. Chakerian
Affiliation: Department of Mathematics, University of California, Davis, California 95616

E. Lutwak
Affiliation: Department of Applied Mathematics and Physics, Polytechnic University, Brooklyn, New York 11201
Email: lutwak@magnus.poly.edu

DOI: https://doi.org/10.1090/S0002-9947-97-01760-1
Keywords: Convex body, mixed volume, quermassintegral, zonoid, generalized zonoid, relative girth, relative brightness
Received by editor(s): October 23, 1995
Additional Notes: Research supported, in part, by NSF Grants DMS–9123571, and DMS–9507988
Article copyright: © Copyright 1997 American Mathematical Society

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