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 $c(K)$.

We give an answer to these questions for $n \geq 5$. In particular, we construct a convex body which has only one section whose centroid coincides with the centroid of the body. Joint work with S. Myroshnychenko and V. Yaskin.