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 â 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. and 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. 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. 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 â 1)-dimensional volume of the intersection of K with the hyperplane u<sup>⊥</sup>. Equivalently, the radial function of the intersection body IK is the Funk transform of the radial function of K. Intersection bodies were introduced by .
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 used this to show that the unit balls l, 2 < p ⤠â in n-dimensional space with the l<sup>p</sup> norm are intersection bodies for n=4 but are not intersection bodies for n âÂÂ¥ 5.