Projection body explained

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 -1} _{n(K)} V ^{\circ} _{n(\Pi} K)\le V ^{n)} _{n(B} ^{n-1} V ^{\circ} _{n(\Pi} 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 -1} _{n(K)} V ^{\circ} _{n(\Pi} K)\ge V ^{n)} _{n(T} ^{n-1} V ^{\circ} _{n(\Pi} 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 .

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, 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.

See also

• Busemann–Petty problem
• Shephard's problem

References

• • • • • • • Schneider. Rolf. Zur einem Problem von Shephard über die Projektionen konvexer Körper. Mathematische Zeitschrift. 101. 1967. 71 - 82. German. 10.1007/BF01135693 . free.