Let $C$ be a convex body in $\mathbb R^n$, i.e a bounded convex subset of $\mathbb R^n$ which has nonempty interior, and which is (A) open, or (B) closed (I'm not sure one makes more sense; choose the one which works best for you). Consider the number $\tau_C \in [0, 1]$ defined by
$$ \tau_C := \frac{1}{\omega_n}\int_{\|x\| \le 1}\frac{vol(C \cap (x + C))}{vol(C)}dx = \frac{1}{\omega_n}\int_{\|x\| \le 1}\mu_C(x + C)dx, $$ where $\omega_n$ is the volume of the unit ball in $\mathbb R^n$ and $\mu_C$ is the uniform measure of $C$. In general, estimating $\tau_C$ would be challenging, since $C$ can be very arbitrary. An exception is when $C$ is euclidean ball. The goal is then to obtain good upper-bounds on $\tau_C$.
The baseline approach. Let $C'$ be the Steiner symmetrization of $C$ w.r.t a coordinate axis. In this post (claim 4; also see comments under the question), it was shown that $\tau_C \le \tau_{C'}$. On the other hand, it is well-known that $C$ can be symmetrized into a ball $B$ of same volume as $C$, via a process $C \rightarrow C' \rightarrow \ldots \rightarrow C^{(k)} \rightarrow \ldots \rightarrow B$ (which is convergent in Hausdorff distance), it follows that $\tau_C \le \tau_B$.
Question
- Is there an alternative way to establish that $\tau_C \le \tau_B$ (where $B$ is a ball of same volume as $C$) without using Steiner symmetrization, but perhaps, using another mapping / process. Can this be done using, for example the Brenier map between $C$ and $B$ ?
- N.B.: Breniers map because, after all, such maps are used to prove the minimality of the, say, the surface of a ball (via Brunn-Minkowski), and so maybe they could also be of help here.
- Is there perhaps a way to directly upper bound $\tau_C$ as a function of its volume ?
Thanks in advance for your help!
