Projection Body

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 $u\in S^$, 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 $K$,
:$V_n(K)^ V_n(\Pi^\circ K)\le V_n(B^n)^ V_n(\Pi^\circ B^n),$
where $B^n$ denotes the ''n''-dimensional unit ball and $V_n$ is ''n''-dimensional volume, and there is equality precisely for ellipsoids. proved that for all convex bodies $K$,
:$V_n(K)^ V_n(\Pi^\circ K)\ge V_n(T^n)^ V_n(\Pi^\circ T^n),$
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 .

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

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*{{Citation , last1=Zhang , first1=Gaoyong , title=Restricted chord projection and affine inequalities , mr=1119653 , year=1991 , journal=Geometriae Dedicata , volume=39 , issue=4 , pages=213–222, doi=10.1007/BF00182294

Convex geometry