On the centre of gravity and width of lattice-constrained convex sets in the plane (Q1368022)

A counterexample is given to the following conjecture of \textit{P. R. Scott} [Am. Math. Mon. 89, 460--461 (1982; Zbl 0499.10034)]. The minimum width \(\omega (K)\) of a convex set \(K\) in the plane having its centroid as the only interior point at an integer lattice point satisfies \(\omega (K) \leq {3\sqrt{2} \over 2}\). The inequality is proved, however, if \(K\) is a triangle. The bound \(\omega (K) \leq \sqrt5\) is conjectured to hold in general.