Problem 1677

Let $C$ denote the $7$-point configuration and let $[C]$ denote its convex hull. The latter is either a triangle formed by three vertices of the quadrangle or the quadrangle itself. Since the diameter of $[C]$ is the longest distance determined by some pair of vertices, it is sufficient to show that this diameter is greater than $2$.

To this end, consider the closed discs of radius $1/2$ centered at those points of $C$ that are not vertices of $[C]$; call these discs, 'inner' discs.

If one of the sides of $[C]$ intersects two inner discs, then its length is at least $3\sqrt{3}/2 > 2$.

Assume henceforth that no side of $[C]$ intersects two inner discs and consider the number of sides of $[C]$ intersecting none of these discs.

If this number is at least two, and $[C]$ is a triangle, then the latter covers at least three inner discs, so its area is at least $3\pi/4$, and its longest side has length at least $\sqrt{3\pi/2} > 2$. And if $[C]$ is a quadrangle, then it covers at least one inner disc, at least half of each of the other two, and the four internal vertex sectors of radius $1/2$; the area of $[C]$ is again greater than or equal to the total area of three discs of radius $1/2$, so its longest diagonal has length at least $\sqrt{3\pi/2} > 2$.

We are left with the case where the number of sides of $[C]$ intersecting no inner disc is at most one.

If $[C]$ is a triangle, refer again to the area argument above: Either $[C]$ covers at least two inner discs and at least half of a third or it covers at least one inner disc and at least half of each of the other three; it also covers the three internal vertex sectors of radius $1/2$, so its area is again greater than or equal to the total area of three discs of radius $1/2$.

Finally, if $[C]$ is a quadrangle, then it has at least three sides of length greater than or equal to $\sqrt{3}$ each. If it is a rectangle, then the length of the diagonal is at least $\sqrt{6} > 2$. Otherwise, a diagonal not passing through the vertex of an obtuse internal angle (there is at least one such) has length greater than $2$.