In convex geometry, the projection body ΠK of a centrally symmetric convex body K in Euclidean space is the convex body such that for any vector u, the support function of ΠK in the direction u is the (n – 1)-dimensional volume of the projection of K onto the hyperplane u⊥.
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. Projection bodies were discussed by Petty (1967), and intersection bodies were introduced by Lutwak (1988).
Minkowski showed that the projection body of a convex body is convex.
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.
- 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, MR 950986, doi:10.1007/BFb0081746
- Koldobsky, Alexander (1998a), "Intersection bodies, positive definite distributions, and the Busemann-Petty problem" (PDF), American Journal of Mathematics, 120 (4): 827–840, ISSN 0002-9327, MR 1637955, doi:10.1353/ajm.1998.0030
- Koldobsky, Alexander (1998b), "Intersection bodies in R⁴", Advances in Mathematics, 136 (1): 1–14, ISSN 0001-8708, MR 1623669, doi:10.1006/aima.1998.1718
- Lutwak, Erwin (1988), "Intersection bodies and dual mixed volumes", Advances in Mathematics, 71 (2): 232–261, ISSN 0001-8708, MR 963487, doi:10.1016/0001-8708(88)90077-1
- Petty, C. M. (1967), "Projection bodies", Proceedings of the Colloquium on Convexity (Copenhagen, 1965), Kobenhavns Univ. Mat. Inst., Copenhagen, pp. 234–241, MR 0216369