Paul Goodey - Inequalities between projection functions of convex bodies

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Paul Goodey - Inequalities between projection functions of convex bodies

	PAUL GOODEY, Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019, USA
	Inequalities between projection functions of convex bodies

(Joint work with Gaoyong Zhang)

A convex body in n-dimensional Euclidean space $\bf E^n$ is a compact convex subset with non-empty interior. It is said to be centrally symmetric if it is a translate of its reflection in the origin. The starting point of our investigations can be found in the work of Shephard [1964] who asked the following question:

If K and L are centrally symmetric convex bodies in $\bf E^n$ , is there the implication

$\begin{displaymath} \textrm{vol}_{n-1}(K\vert u^\bot) \ge \textrm{vol}_{n-1}(L\v... ...\in S^{n-1} \Rightarrow \textrm{vol}_n(K)\ge \textrm{vol}_n(L)?\end{displaymath}$

Here, $K\vert u^\bot$ is the orthogonal projection of K onto the subspace of $\bf E^n$ orthogonal to the unit vector $u\in S^{n-1}$ . The question was answered independently by Petty [1967] and Schneider [1967]. They both showed that the answer is affirmative if it is further assumed that K is a zonoid. Both authors also showed that this further assumption cannot be suppressed.

The generalization of the Shephard problem for lower dimensional projections of convex bodies has been open since Petty and Schneider's work, see question 4.2.1 of Gardner [1995]. It states:

If K and L are centrally symmetric convex bodies in $\bf E^n$ , and $i\in\{1,\dots,n-1\}$ , is there the implication

$\begin{displaymath} \textrm{vol}_i(K\vert E)\ge \textrm{vol}_i(L\vert E) \quad \... ...cal L}_i^n \Rightarrow \textrm{vol}_n(K)\ge \textrm{vol}_n(L)?\end{displaymath}$

Here ${\cal L}_i^n$ denotes the Grassmann manifold of all i-dimensional subspaces of $\bf E^n$ . The counterpart of this problem for cross sections of convex bodies is considered by Bourgain and Zhang [1996], and by Zhang [1996].

We will obtain an answer to this generalization of Shephard's problem which is analogous to the results of Petty and Schneider. This will be achieved by establishing injectivity results for various integral transforms on Grassmann manifolds. These transforms arise very naturally in the geometry of convex bodies and their injectivity properties have special implications for bodies of revolution.

Next: Eric L. Grinberg - Up: Convex Geometry / Géométrie Previous: Richard J. Gardner -

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