Projection body

In convex geometry, the projection body [math]\displaystyle{ \Pi K }[/math] of a convex body [math]\displaystyle{ K }[/math] in n-dimensional Euclidean space is the convex body such that for any vector [math]\displaystyle{ u\in S^{n-1} }[/math], the support function of [math]\displaystyle{ \Pi K }[/math] in the direction u is the (n – 1)-dimensional volume of the projection of K onto the hyperplane orthogonal to u. Hermann 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 [math]\displaystyle{ K }[/math] a convex body, let [math]\displaystyle{ \Pi^\circ K }[/math] 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 [math]\displaystyle{ K }[/math],

[math]\displaystyle{ V_n(K)^{n-1} V_n(\Pi^\circ K)\le V_n(B^n)^{n-1} V_n(\Pi^\circ B^n), }[/math]

where [math]\displaystyle{ B^n }[/math] denotes the n-dimensional unit ball and [math]\displaystyle{ V_n }[/math] is n-dimensional volume, and there is equality precisely for ellipsoids. Zhang (1991) proved that for all convex bodies [math]\displaystyle{ K }[/math],

[math]\displaystyle{ V_n(K)^{n-1} V_n(\Pi^\circ K)\ge V_n(T^n)^{n-1} V_n(\Pi^\circ T^n), }[/math]

where [math]\displaystyle{ T^n }[/math] denotes any [math]\displaystyle{ n }[/math]-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 lpn, 2 < p ≤ ∞ in n-dimensional space with the l^p norm are intersection bodies for n=4 but are not intersection bodies for n ≥ 5.

See also

• Busemann–Petty problem
• Shephard's problem

References

• 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, ISBN 978-3-540-19353-1 
• Koldobsky, Alexander (1998a), "Intersection bodies, positive definite distributions, and the Busemann-Petty problem", American Journal of Mathematics 120 (4): 827–840, doi:10.1353/ajm.1998.0030, ISSN 0002-9327 
• Koldobsky, Alexander (1998b), "Intersection bodies in R⁴", Advances in Mathematics 136 (1): 1–14, doi:10.1006/aima.1998.1718, ISSN 0001-8708 
• 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 
• Petty, Clinton M. (1967), "Projection bodies", Proceedings of the Colloquium on Convexity (Copenhagen, 1965), Kobenhavns Univ. Mat. Inst., Copenhagen, pp. 234–241, https://books.google.com/books?id=UIjyTgEACAAJ 
• 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 
• Schneider, Rolf (1967). "Zur einem Problem von Shephard über die Projektionen konvexer Körper" (in German). Mathematische Zeitschrift 101: 71–82. doi:10.1007/BF01135693. 
• Zhang, Gaoyong (1991), "Restricted chord projection and affine inequalities", Geometriae Dedicata 39 (4): 213–222, doi:10.1007/BF00182294