Projection body

In convex geometry, the projection body $Π K$ 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 $Π K$ 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. Template:Harvtxt and Template:Harvs used projection bodies in their solution to Shephard's problem.

For $K$ a convex body, let $Π^{\circ} K$ denote the polar body of its projection body. There are two remarkable affine isoperimetric inequality for this body. Template:Harvtxt proved that for all convex bodies $K$ ,

V_{n} (K)^{n - 1} V_{n} (Π^{\circ} K) \leq V_{n} (B^{n})^{n - 1} V_{n} (Π^{\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. Template:Harvs proved that for all convex bodies $K$ ,

V_{n} (K)^{n - 1} V_{n} (Π^{\circ} K) \geq V_{n} (T^{n})^{n - 1} V_{n} (Π^{\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^⊥. Equivalently, the radial function of the intersection body IK is the Funk transform of the radial function of K. Intersection bodies were introduced by Template:Harvs.

Template:Harvtxt 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 Template:Harvtxt used this to show that the unit balls lTemplate:Su, 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.

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Projection body

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Projection body

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