Projection body

In convex geometry, the projection body $$\Pi K$$ of a convex body $$K$$ in n-dimensional Euclidean space is the convex body such that for any vector $$u\in S^{n-1}$$, the support function of $$\Pi K$$ 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. and used projection bodies in their solution to Shephard's problem.

For $$K$$ a convex body, let $$\Pi^\circ K$$ denote the polar body of its projection body. There are two remarkable affine isoperimetric inequality for this body. proved that for all convex bodies $$K$$,
 * $$ V_n(K)^{n-1} V_n(\Pi^\circ K)\le V_n(B^n)^{n-1} V_n(\Pi^\circ B^n),$$

where $$B^n$$ denotes the n-dimensional unit ball and $$V_n$$ is n-dimensional volume, and there is equality precisely for ellipsoids. proved that for all convex bodies $$K$$,
 * $$ V_n(K)^{n-1} V_n(\Pi^\circ K)\ge V_n(T^n)^{n-1} V_n(\Pi^\circ T^n),$$

where $$T^n$$ denotes any $$n$$-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&perp;. 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$p n$, 2 < p ≤ ∞ in n-dimensional space with the lp norm are intersection bodies for n=4 but are not intersection bodies for n ≥ 5.