Speaker: Alexander Magazinov (TAU) Title: The depth of the centerline in R^d Abstract: --------- Let $\mu$ be a probability measure in $\mathbb R^d$. Rado's Centerpoint Theorem (1946) says that there exists a "centerpoint" $o \in \mathbb R^d$ such that every closed half-space $H$ with $o \in H$ satisfies $\mu(H) \geq 1 / (d + 1)$. If we replace a point by a line, a natural question arises: What is the maximal $f(d)$ for which we can find a (1-dimensional) line $\ell \subset \mathbb R^d$ such that $\mu(H) \geq f(d)$ whenever $H$ is a closed subspace with $\ell \subset H$? Projecting along any 1-dimensional direction and applying Rado's theorem shows that $f(d) = 1 / d$ is sufficient. I will show a slight improvement on this "trivial" bound to $f(d) = 1/d + 1/(3d^3)$, which is, according to B. Bukh, the first such improvement. The conjectured bound is $f(d) = 2/(d + 1)$, see (Bukh - Matousek - Nivasch, 2008) for the conjecture and (Bukh, Nivasch, 2012) for the fact that $2/(d + 1)$ cannot be improved further. The talk is based on a joint work with Attila Por.