How far apart can the projection of the centroid of a convex body and the centroid of its projection be? (Q6624758)

For a hyperplane \(H\subset{\mathbb R}^n\), let \(P_H\) denote the orthogonal projection to \(H\), and for a convex body \(K\subset{\mathbb R}^n\), let \(c(K)\) be its centroid. The authors are interested in the smallest constant \(D_n\) such that for any convex body \(K\subset{\mathbb R}^n\) one has \N\[\N|P_H c(K) -c(P_HK)|\le D_nw_K(u),\N\]\Nwhere \(u\) is a unit vector parallel to \(P_H c(K) -c(P_HK)\) (if this is \(\not= o\)) and \(w_K(u)\) is the width of \(K\) in direction \(u\). It is known that \(D_2=1/6\). The authors prove that \(D_3=1-\sqrt{2/3}\), the sequence \((D_n)_{n=3}^\infty\) is increasing, and \(\lim_{n\to\infty} \approx 0.2016\). Moreover, they describe the convex bodies achieving the extreme values \(D_n\). The proof is in principle elementary, though elaborate. The paper concludes with remarks on projections onto lower-dimensional subspaces.