Projection body

In convex geometry, the projection body ${\displaystyle \Pi K}$ of a convex body ${\displaystyle K}$ in n-dimensional Euclidean space is the convex body such that for any vector ${\displaystyle u\in S^{n-1}}$, the support function of ${\displaystyle \Pi K}$ 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 ${\displaystyle K}$ a convex body, let ${\displaystyle \Pi ^{\circ }K}$ 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 ${\displaystyle K}$,

${\displaystyle V_{n}(K)^{n-1}V_{n}(\Pi ^{\circ }K)\leq V_{n}(B^{n})^{n-1}V_{n}(\Pi ^{\circ }B^{n}),}$

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

${\displaystyle V_{n}(K)^{n-1}V_{n}(\Pi ^{\circ }K)\geq V_{n}(T^{n})^{n-1}V_{n}(\Pi ^{\circ }T^{n}),}$

where ${\displaystyle T^{n}}$ denotes any ${\displaystyle n}$-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 l^p
_n, 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.