An origin-symmetric convex body $K$ in $\mathbb{R}^n$ is said to be in the John position if the maximal volume ellipsoid contained in it is the Euclidean ball.

In 1961, Grunbaum asked whether the centroid $c(K)$ of a convex body $K$ is the centroid of at least $n + 1$ different $(n − 1)$-dimensional sections of $K$ through $c(K)$. A few years later, Loewner asked to find the minimum number of hyperplane sections of $K$ passing through $c(K)$ whose centroid is the same as $c(K)$.