Lower Bounds on the Dilation of Plane Spanners

(I) We exhibit a set of 23 points in the plane that has dilation at least 1.4308, improving the previously best lower bound of 1.4161 for the worst-case dilation of plane spanners. (II) For every n >= 13, there exists an n-element point set S such that the degree 3 dilation of S denoted by delta(0)(S, 3) equals 1+root 3 = 2.7321... in the domain of plane geometric spanners. In the same domain, we show that for every n >= 6, there exists a an n-element point set S such that the degree 4 dilation of S denoted by delta(0)(S, 4) equals 1+ root(5-root 5)/2 = 2.1755... The previous best lower bound of 1.4161 holds for any degree. (III) For every n >= 6, there exists an n-element point set S such that the stretch factor of the greedy triangulation of S is at least 2.0268.