On the intersection of random rotations of a symmetric convex body (Q2877359)

Let \(C \subset \mathbb{R}^n\) be a symmetric convex body of volume 1 and let \(U \in O(n)\) be an orthogonal transformation. The authors study the expected volume of the intersection of the convex body with its image under a random orthogonal transformation. More precisely, they are interested in bounding the integral NEWLINE\[NEWLINE \int_{O(n)}\mathrm{vol}(C \cap U(C) ) \;d\mu(U), NEWLINE\]NEWLINE in which \(\mu\) is the Haar measure on \(O(n)\). Using different natural assumptions on the allowed bodies \(C\) they derive non-trivial upper bounds (Theorem 1 and 2) for this integral. Theorem 3 addresses lower bounds, whereas the last section of the paper applies these results to \(L_q\)-centroid bodies of an isotropic log-concave measure on \(\mathbb{R}^n\).