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Projection body

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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 u\in 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.

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 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} ,

V n ( K ) n − 1 V n ( Π ∘ K ) ≤ V n ( B n ) n − 1 V n ( Π ∘ B n ) , {\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 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} ,

V n ( K ) n − 1 V n ( Π ∘ K ) ≥ V n ( T n ) n − 1 V n ( Π ∘ T n ) , {\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 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 l
n, 2 < p ≤ ∞ in n-dimensional space with the l norm are intersection bodies for n=4 but are not intersection bodies for n ≥ 5.

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