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, Lowner asked to find the minimum number of hyperplane section of K passing through c(K) whose centroid is the same as c(K).

We give an answer to these questions for n ≥ 5. In particular, we construct a convex body which has only one section whose centroid coincides with the centroid of the body by using Fourier analytic tools and exploiting the existence of non-intersection bodies in these dimensions. Joint work with S. Myroshnychenko and V. Yaskin.