Projection body  In convex geometry, the projection body formula_1 of a convex body formula_2 in "n"-dimensional Euclidean space is the convex body such that for any vector formula_3, the support function of formula_1 in the direction "u" is the ("n" – 1)-dimensional volume of the projection of "K" onto the hyperplane orthogonal to "u". Minkowski showed that the projection body of a convex body is convex. and used projection bodies in their solution to Shephard's problem. For formula_2 a convex body, let formula_6 denote the polar body of its projection body. There are two remarkable affine isoperimetric inequality for this body. proved that for all convex bodies formula_2, where formula_9 denotes the "n"-dimensional unit ball and formula_10 is "n"-dimensional volume, and there is equality precisely for ellipsoids. proved that for all convex bodies formula_2, where formula_13 denotes any formula_14-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 .