Projection body

In convex geometry, the projection body Π K {displaystyle Pi K} of a convex body K {displaystyle K} in n-dimensional Euclidean space is the convex body such that for any vector u ∈ S n − 1 {displaystyle uin S^{n-1}} , the support function of Π K {displaystyle Pi K} in the direction u is the (n – 1)-dimensional volume of the projection of K onto the hyperplane orthogonal to u. In convex geometry, the projection body Π K {displaystyle Pi K} of a convex body K {displaystyle K} in n-dimensional Euclidean space is the convex body such that for any vector u ∈ S n − 1 {displaystyle uin S^{n-1}} , the support function of Π K {displaystyle Pi K} 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 K {displaystyle K} a convex body, let Π ∘ K {displaystyle Pi ^{circ }K} 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 K {displaystyle K} , where B n {displaystyle B^{n}} denotes the n-dimensional unit ball and V n {displaystyle V_{n}} is n-dimensional volume, and there is equality precisely for ellipsoids. Zhang (1991) proved that for all convex bodies K {displaystyle K} , where T n {displaystyle T^{n}} denotes any n {displaystyle n} -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 lp norm are intersection bodies for n=4 but are not intersection bodies for n ≥ 5.