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Transactions of the American Mathematical Society

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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
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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  • [1937] A. D. Aleksandrov, On the theory of mixed volumes. II. New inequalities between mixed volumes and their application, Mat. Sbornik N.S. 2 (1937), 1205-1238, Russian.
  • [1988] Yu. D. Burago and V. A. Zalgaller, Geometric Inequalities, Springer Verlag, Heidelberg, 1988. MR 89b:52020
  • [1967] G. D. Chakerian, Sets of constant relative width and constant relative brightness, Trans. Amer. Math. Soc. 129 (1967), 26-37.MR 35:3545
  • [1979] V. P. Fedotov, A counterexample to a hypothesis of Firey, Math. Zametki 26 (1979), 269-275, Russian. MR 80i:52009
  • [1970] Wm. J. Firey, Convex bodies of constant outer $p$-measure, Mathematika 17 (1970), 21-27. MR 42:2367
  • [1980] P. R. Goodey and R. Schneider, On intermediate area functions of convex bodies, Math. Z. 173 (1980), 185-194. MR 81k:52010
  • [1993] P. R. Goodey and W. Weil, Zonoids and generalizations, Handbook of Convex Geometry (P.M. Gruber and J.M. Wills, Eds.), North-Holland, Amsterdam, 1993. MR 95g:52015
  • [1980] K. Leichtweiß, Konvexe Mengen, Springer, Berlin, 1980. MR 81j:52001
  • [1967] C. M. Petty, Projection bodies, Proc. Coll. Convexity, Copenhagen, 1965, Københavns Univ. Mat. Inst., 1967, pp. 234-241. MR 35:7203
  • [1967] R. Schneider, Zur einem Problem von Shephard über die Projektionen konvexer Körper, Math. Z. 101 (1967), 71-82. MR 36:2059
  • [1985] R. Schneider, On the Aleksandrov-Fenchel inequality, Ann. N.Y. Acad. Sci. 440 (1985), 132-141. MR 87c:52019
  • [1990] R. Schneider, On the Aleksandrov-Fenchel inequality for convex bodies, Results in Math. 17 (1990), 287-295. MR 91b:52009
  • [1993] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge U. Press, Cambridge, 1993. MR 94d:52007
  • [1983] R. Schneider and W. Weil, Zonoids and related topics, Convexity and its Applications (P.M. Gruber and J.M. Wills, Eds.), Birkhäuser, Basel, 1983, pp. 296-317. MR 85c:52010
  • [1974] W. Weil, Über den Vektorraum der Differenzen von Stützfunktionen konvexer Körper, Math. Nachr. 59 (1974), 353-369. MR 49:6033
  • [1976] W. Weil, Kontinuierliche Linearkombination von Strecken, Math. Z. 148 (1976), 71-84. MR 53:3887
  • [1979] W. Weil, Centrally symmetric convex bodies and distributions II, Israel J. Math. 32 (1979), 143-182. MR 80g:52003

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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

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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