Projection body

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In convex geometry, the projection body of a convex body in n-dimensional Euclidean space is the convex body such that for any vector , the support function of in the direction u is the (n – 1)-dimensional volume of the projection of K onto the hyperplane orthogonal to u.

Minkowski showed that the projection body of a convex body is convex. Petty (1967) and Schneider (1967) used projection bodies in their solution to Shephard's problem.

For a convex body, let denote the polar body of its projection body. There are two remarkable affine isoperimetric inequality for this body. Petty (1971) proved that for all convex bodies ,

where denotes the n-dimensional unit ball and is n-dimensional volume, and there is equality precisely for ellipsoids. Zhang (1991) proved that for all convex bodies ,

where denotes any -dimensional simplex, and there is equality precisely for such simplices.

The intersection body IK of K is defined similarly, as the star body such that for any vector u the radial function of IK from the origin in direction u is the (n – 1)-dimensional volume of the intersection of K with the hyperplane u. Equivalently, the radial function of the intersection body IK is the Funk transform of the radial function of K. Intersection bodies were introduced by Lutwak (1988).

Koldobsky (1998a) showed that a centrally symmetric star-shaped body is an intersection body if and only if the function 1/||x|| is a positive definite distribution, where ||x|| is the homogeneous function of degree 1 that is 1 on the boundary of the body, and Koldobsky (1998b) used this to show that the unit balls lp
n
, 2 < p ≤ ∞ in n-dimensional space with the lp norm are intersection bodies for n=4 but are not intersection bodies for n ≥ 5.

See also[edit]

References[edit]

  • Bourgain, Jean; Lindenstrauss, J. (1988), "Projection bodies", Geometric aspects of functional analysis (1986/87), Lecture Notes in Math., 1317, Berlin, New York: Springer-Verlag, pp. 250–270, doi:10.1007/BFb0081746, MR 0950986
  • Koldobsky, Alexander (1998a), "Intersection bodies, positive definite distributions, and the Busemann-Petty problem" (PDF), American Journal of Mathematics, 120 (4): 827–840, doi:10.1353/ajm.1998.0030, ISSN 0002-9327, MR 1637955
  • Koldobsky, Alexander (1998b), "Intersection bodies in R⁴", Advances in Mathematics, 136 (1): 1–14, doi:10.1006/aima.1998.1718, ISSN 0001-8708, MR 1623669
  • Lutwak, Erwin (1988), "Intersection bodies and dual mixed volumes", Advances in Mathematics, 71 (2): 232–261, doi:10.1016/0001-8708(88)90077-1, ISSN 0001-8708, MR 0963487
  • Petty, Clinton M. (1967), "Projection bodies", Proceedings of the Colloquium on Convexity (Copenhagen, 1965), Kobenhavns Univ. Mat. Inst., Copenhagen, pp. 234–241, MR 0216369
  • Petty, Clinton M. (1971), "Isoperimetric problems", Proceedings of the Conference on Convexity and Combinatorial Geometry (Univ. Oklahoma, Norman, Okla., 1971). Dept. Math., Univ. Oklahoma, Norman, Oklahoma, pp. 26–41, MR 0362057
  • Schneider, Rolf (1967). "Zur einem Problem von Shephard über die Projektionen konvexer Körper". Math. Z. (in German). 101: 71&ndash, 82. doi:10.1007/BF01135693.
  • Zhang, Gaoyong (1991), "Restricted chord projection and affine inequalities", Geometriae Dedicata, 39 (4): 213–222, MR 1119653