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  • Print publication year: 2006
  • Online publication date: June 2013

4 - Projection bodies and volume inequalities

Summary

In the first section of this chapter, we meet projection bodies. A projection body is a centered convex body whose support function gives the volumes of projections on hyperplanes of another convex body. In Theorem 4.1.11, we see that projection bodies are just centered zonoids. Every ellipsoid is a zonoid; moreover, Corollary 4.1.12 characterizes zonoids as Hausdorff limits of finite Minkowski sums of ellipsoids. Zonoids are “supersymmetric” convex bodies, of proven and wide applicability in mathematics and other subjects. Particularly susceptible to analytical methods, zonoids have found a prominent place not only in geometry, but also in the local theory of Banach spaces. Generalized zonoids, which constitute a stepping-stone from zonoids to general centrally symmetric convex bodies, are also introduced. Here some properties of the cosine transform, expounded in Appendix C, come into play.

Section 4.2 deals with a question, sometimes called Shephard's problem, of comparative information: If the area of the shadow on a plane of one convex body is always less than that of another, is the volume of the one body less than that of the other? The solution employs the Brunn–Minkowski theory and zonoids. Theorem 4.2.4 exhibits a pair of centrally symmetric convex bodies for which the answer is negative. If the body whose shadows have larger area is a zonoid, however, the answer is positive, as Corollary 4.2.7 demonstrates.

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