Intersection body

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In convex geometry, the projection body Π K {\displaystyle \Pi K} {\displaystyle \Pi K} of a convex body K {\displaystyle 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}} {\displaystyle u\in S^{n-1}}, the support function of Π K {\displaystyle \Pi 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} {\displaystyle K} a convex body, let Π ∘ K {\displaystyle \Pi ^{\circ }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} {\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}),} {\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}} {\displaystyle B^{n}} denotes the n-dimensional unit ball and V n {\displaystyle 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} {\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}),} {\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}} {\displaystyle T^{n}} denotes any n {\displaystyle 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 lp
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.

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